2413:
11736:
13944:
19616:
2233:
18116:
4783:
15114:
4462:
5955:
11591:
10122:
5633:
11067:
12552:
8617:
17392:
9638:
17923:
11892:
11434:
14277:
16156:
10335:
9872:
19002:
16163:
Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between
9399:
3938:
16348:
15479:
6312:
13783:
2408:{\displaystyle {\begin{bmatrix}1\\&1\\&&\ddots \\&&&1\\a_{1,1}&\cdots &\cdots &a_{1,k}\\\vdots &&&\vdots \\a_{n-k,1}&\cdots &\cdots &a_{n-k,k}\end{bmatrix}}}
19491:
15965:
1170:, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is
10570:
19815:
18029:
14573:
gives rise to multiplication of the Plücker coordinates by a nonzero constant (the determinant of the change of basis matrix), these are only defined up to projective equivalence, and hence determine a point in
12768:
4881:
11288:
19980:, which is the enumerative geometry involved in calculating the number of points, lines, planes, etc. in a projective space that intersect a given set of points, lines, etc., using the intersection theory of
19480:
4583:
5167:
4166:
17038:
16789:
11513:
15196:
10217:
9734:
6448:
17153:
16903:
14764:
14690:
8106:
7540:
6706:
15577:
15671:
12420:
12103:
17689:
16975:
14925:
7226:
19274:
17765:
14007:
11803:
8768:
7310:
2890:
2650:
17561:
11164:
19421:
18179:
17611:
17494:
11215:
10666:
4282:
950:
419:
19114:
12248:
12032:
11731:{\displaystyle \mathbf {P} ({\mathcal {G}})\to \mathbf {P} \left({\mathcal {E}}_{\mathbf {Gr} (k,{\mathcal {E}})}\right)=\mathbf {P} ({\mathcal {E}})\times _{S}\mathbf {Gr} (k,{\mathcal {E}}).}
5819:
7836:
7474:
7024:
6128:
6079:
18339:
These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of
12349:
11577:
10737:
5016:
19933:
9947:
19953:. Under the Cartan embedding, their connected components are equivariantly diffeomorphic to the projectivized minimal spinor orbit, under the spin representation, the so-called projective
18897:
18588:
5259:
15726:
14874:
3364:
15767:
14613:
13366:
19736:
5534:
18850:
17200:
14328:
7970:
7402:
3535:
2990:
1825:
1591:
7122:
6928:
6224:
6189:
16839:
13654:
8299:
5725:
16554:
14080:
12435:
8478:
7609:
6749:
5069:
4573:
4520:
4272:
4219:
4074:
3755:
3698:
3645:
3588:
3300:
3209:
3089:
2943:
2729:
15248:
14531:
14380:
13596:
13458:
12940:
8387:
8022:
7888:
7367:
2781:
20604:
Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator
Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–".
18740:
16720:
16507:
14915:
13753:
13325:
5472:
5378:
5208:
4021:
3416:
1540:
1465:
1378:
1255:
556:
475:
214:
72:
19205:
13263:
6975:
16419:
12612:
11959:
10913:
10160:
9677:
9225:
8697:
6490:
6359:
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5995:
5769:
1151:
1077:
to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing
596:
515:
7794:
6614:
20015:. A similar construction holds for solutions of the BKP integrable hierarchy, in terms of abelian group flows on an infinite dimensional maximal isotropic Grassmann manifold.
19371:
17247:
9018:
8941:
8876:
8654:
8154:
7171:
7073:
872:
843:
764:
12305:
1868:
268:
20007:. These can be expressed in terms of abelian group flows on an infinite-dimensional Grassmann manifold. The KP equations, expressed in Hirota bilinear form in terms of the
12175:
11458:
11352:
10968:
10829:
10765:
10409:
9435:
9278:
8233:
7728:
19307:
17971:
15338:
14479:
12665:
9490:
3813:
3247:
2456:
9087:
9065:
9043:
8842:
8820:
2179:
1917:
1778:
1729:
20114:
20104:
19940:
18767:
18481:
17444:
13059:
20004:
19861:
17775:
16379:
15821:
13504:
12799:
8439:
8344:
5524:
5408:
1424:
19154:
17237:
17084:
16019:
15379:
15308:
14429:
14106:
13187:
12989:
12888:
7566:
6640:
6154:
4907:
3969:
3442:
3036:
2676:
2091:
1996:
1950:
1617:
2512:
10958:
7669:
6868:
6809:
20377:
Harnad, J.; Shnider, S. (1992). "Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions".
13217:
13138:
9939:
9165:
976:
790:
358:
18282:
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15275:
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3136:
1691:
1326:
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11357:
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15405:
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14790:
12274:
12131:
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18401:
18377:
18357:
18334:
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18199:
18015:
17991:
17709:
17524:
16621:
16574:
16439:
16202:
16182:
15904:
15795:
14571:
14551:
14449:
14400:
14027:
13773:
13701:
13544:
13524:
13478:
13406:
13386:
13158:
13110:
12960:
12859:
12839:
12819:
12685:
12632:
12195:
12151:
12052:
11979:
11533:
11112:
11092:
10933:
10873:
10849:
10805:
10785:
10693:
10615:
10595:
10493:
10473:
10453:
10433:
9901:
9754:
9482:
9245:
9185:
9127:
9107:
8980:
8907:
8795:
8407:
8253:
8200:
8180:
7935:
7750:
7691:
7632:
7422:
7142:
7044:
6890:
6831:
6772:
6542:
6510:
5809:
5789:
5678:
5658:
5497:
5428:
5339:
5319:
5299:
5279:
4927:
3775:
3462:
3156:
3109:
3010:
2829:
2801:
2592:
2572:
2552:
2532:
2476:
2223:
2203:
2135:
2111:
2036:
2016:
1970:
1749:
1664:
1644:
1485:
1398:
1299:
1279:
1067:
1047:
1016:
996:
896:
814:
735:
715:
656:
636:
616:
308:
288:
234:
173:
150:
127:
100:
19313:
with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.
14177:
14172:
13087:
3488:
2062:
16082:
10225:
9762:
18907:
20034:
15407:(the simplest Grassmannian that is not a projective space), the above reduces to a single equation. Denoting the homogeneous coordinates of the image
9286:
3823:
16271:
15410:
13939:{\displaystyle W^{T}=={\begin{bmatrix}w_{11}&\cdots &w_{1n}\\\vdots &\ddots &\vdots \\w_{k1}&\cdots &w_{kn}\end{bmatrix}},}
6232:
19999:
A further application is to the solution of hierarchies of classical completely integrable systems of partial differential equations, such as the
19988:
can also be used to parametrize simultaneous eigenvectors of complete sets of commuting operators in quantum integrable spin systems, such as the
19611:{\displaystyle {\widetilde {\mathbf {Gr} }}_{k}(\mathbf {R} ^{n})=\operatorname {SO} (n)/(\operatorname {SO} (k)\times \operatorname {SO} (n-k)).}
18111:{\displaystyle \chi _{k,n}={\begin{pmatrix}\left\lfloor {\frac {n}{2}}\right\rfloor \\\left\lfloor {\frac {k}{2}}\right\rfloor \end{pmatrix}}}
15918:
21002:
10507:
20250:
20220:
19741:
21032:
4778:{\displaystyle {\hat {A}}^{j_{1},\dots ,j_{k}}={\hat {A}}^{i_{1},\dots ,i_{k}}({\hat {A}}_{j_{1},\dots ,j_{k}}^{i_{1},\dots ,i_{k}})^{-1},}
16603:
is equivalent to the choice of an inner product, so with respect to the chosen inner product, this isomorphism of
Grassmannians sends any
12695:
6547:
For the case of real or complex
Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.
4793:
11220:
19426:
5074:
4079:
16988:
16729:
11463:
20094:
are generalizations of
Grassmannians whose elements, viewed geometrically, are nested sequences of subspaces of specified dimensions.
15124:
10165:
9682:
17089:
16850:
14695:
14621:
8032:
7479:
6645:
20912:
15484:
15587:
15109:{\displaystyle \sum _{l=1}^{k+1}(-1)^{\ell }w_{i_{1},\dots ,i_{k-1},j_{l}}w_{j_{1},\dots ,{\widehat {j_{l}}},\dots j_{k+1}}=0,}
12359:
12057:
8948:
21340:
21311:
21243:
21214:
20998:
20353:
20234:
19973:
A key application of
Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.
17632:
16916:
7176:
4457:{\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}W_{i_{1},\dots ,i_{k}}={\hat {A}}^{j_{1},\dots ,j_{k}}W_{j_{1},\dots ,j_{k}},}
20740:"Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifolds (Random Systems and Dynamical Systems)"
20048:
tasks of video-based face recognition and shape recognition, and are used in the data-visualization technique known as the
19210:
17714:
13954:
11755:
8705:
7236:
5950:{\displaystyle \mathbf {Gr} (k,V)\sim \left\{P\in \mathrm {End} (V)\mid P=P^{2}=P^{\dagger },\,\mathrm {tr} (P)=k\right\}.}
2837:
2597:
17529:
11117:
6373:
20000:
19376:
18134:
17566:
17449:
11169:
10740:
10620:
905:
374:
20813:
Chakravarty, S.; Kodama, Y. (July 2009). "Soliton
Solutions of the KP Equation and Application to Shallow Water Waves".
19046:
12200:
11984:
20414:"Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces"
10117:{\displaystyle \mathbf {Gr} (k,V)=U(V,h)/\left(U(w_{0},h|_{w_{0}})\times U(w_{0}^{\perp }|,h_{w_{0}^{\perp }})\right),}
7807:
7430:
6980:
6084:
6035:
21275:
21195:
21173:
21113:
20789:
20714:
20283:
20008:
12310:
11538:
10698:
4959:
19871:
21295:
21235:
18855:
18504:
6524:. The exact inner product used does not matter, because a different inner product will give an equivalent norm on
5628:{\displaystyle P_{w}(v)={\begin{cases}v\quad {\text{ if }}v\in w\\0\quad {\text{ if }}v\in w^{\perp }.\end{cases}}}
5232:
15681:
14829:
3305:
20049:
15731:
14577:
13330:
12994:
19689:
21330:
18808:
17158:
14286:
12547:{\displaystyle \left\{(x,v)\in \mathbf {P} (V)(K)\times \mathbf {Gr} (k,{\mathcal {E}})(K)\mid x\in v\right\}.}
8612:{\displaystyle H:=\mathrm {stab} (w_{0}):=\{h\in \mathrm {GL} (V)\,|\,h(w_{0})=w_{0}\}\subset \mathrm {GL} (V)}
7940:
7372:
5218:
An alternative way to define a real or complex
Grassmannian as a manifold is to view it as a set of orthogonal
3493:
2948:
1783:
1549:
18496:. The generators are identical to those of the classical cohomology ring, but the top relation is changed to
7078:
6895:
6194:
6159:
21231:
16794:
13601:
8258:
5683:
20:
16512:
14823:
14819:
14032:
13281:
7571:
6711:
5021:
4525:
4472:
4224:
4171:
4026:
3707:
3650:
3597:
3540:
3252:
3161:
3041:
2895:
2681:
21335:
18380:
15201:
14484:
14333:
13549:
13411:
12893:
8352:
7975:
7841:
7320:
2734:
361:
18704:
16684:
16471:
14879:
13706:
13289:
5438:
5344:
5172:
3974:
3369:
1493:
1429:
1331:
1219:
520:
439:
178:
36:
21345:
20858:
Kodama, Yuji; Williams, Lauren (December 2014). "KP solitons and total positivity for the
Grassmannian".
19159:
17387:{\displaystyle X_{\lambda }(k,n)=\{W\in \mathbf {Gr} _{k}(V)\,|\,\dim(W\cap V_{n-k+j-\lambda _{j}})=j\}.}
13227:
6937:
5219:
18336:. In particular, all of the integral cohomology is at even degree as in the case of a projective space.
16384:
13408:
and give another method for constructing the
Grassmannian. To state the Plücker relations, fix a basis
12577:
11933:
10878:
10130:
9647:
9190:
8662:
6324:
6000:
5965:
5734:
1121:
561:
480:
21097:
20267:
20133:
11062:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})\times _{S}S'\simeq \mathbf {Gr} (k,{\mathcal {E}}_{S'})}
8959:
7755:
6582:
19347:
13775:-component column vectors forming the transpose of the corresponding homogeneous coordinate matrix:
13112:
will give a different wedge product, but the two will differ only by a non-zero scalar multiple (the
9633:{\displaystyle \mathbf {Gr} (k,V)=O(V,q)/\left(O(w,q|_{w})\times O(w^{\perp },q|_{w^{\perp }})\right)}
8989:
8912:
8847:
8625:
8125:
7147:
7049:
848:
819:
740:
17203:
12279:
1835:
242:
12156:
11439:
11333:
10810:
10746:
10390:
9408:
9250:
8205:
7700:
5565:
20103:
Given a distinguished class of subspaces, one can define
Grassmannians of these subspaces, such as
19279:
17937:
17918:{\displaystyle \chi _{k,n}=\chi _{k-1,n-1}+(-1)^{k}\chi _{k,n-1},\qquad \chi _{0,n}=\chi _{n,n}=1.}
15313:
14454:
12640:
6458:
3780:
3214:
2423:
103:
12635:
9070:
9048:
9026:
8825:
8803:
8442:
2140:
1878:
1754:
1696:
20119:
20108:
18745:
18439:
17416:
1543:
79:
20085:
19840:
16358:
15800:
13483:
12778:
8412:
8307:
5502:
5387:
1403:
20587:
M. Sato, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds",
20216:
19124:
17209:
17050:
15983:
15346:
15284:
14405:
14085:
13163:
12965:
12864:
7545:
6619:
6133:
4886:
3948:
3421:
3015:
2655:
2070:
1975:
1929:
1596:
1258:
875:
21024:
20012:
2481:
422:
20776:. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press.
20701:. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press.
20067:
16723:
16656:
15977:
15856:
14818:
positive integers, respectively, the following homogeneous quadratic equations, known as the
11887:{\displaystyle \mathbf {P} ({\mathcal {G}}_{T})\to \mathbf {P} ({\mathcal {E}})\times _{S}T.}
11429:{\displaystyle \mathbf {Gr} \left(k,{\mathcal {E}}_{\mathbf {Gr} (k,{\mathcal {E}})}\right),}
10376:
7641:
6840:
6781:
683:
671:
371:, who studied the set of projective lines in real projective 3-space, which is equivalent to
20991:
Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision
13196:
13123:
9909:
9135:
955:
769:
328:
21289:
21123:
21105:
21061:
20956:
20877:
20613:
20425:
20386:
20363:
20293:
20275:
20026:
20022:
solutions of KP equations which are nonsingular for real values of the KP flow parameters.
18596:
18255:
18018:
17994:
17411:
16678:
16626:
16579:
16444:
16239:
16207:
16051:
16024:
15907:
15826:
15253:
13659:
11906:
10412:
10372:
9440:
8448:
7893:
4932:
3114:
1920:
1669:
1624:
1487:
1304:
21168:. Annals of Mathematics Studies. Vol. 76. Princeton, NJ: Princeton University Press.
21143:
21131:
21006:
20993:, CVPR 23–28 June 2008, IEEE Conference on Computer Vision and Pattern Recognition, 2008,
20301:
20081:
19012:
18779:
13277:
12571:
12565:
11296:
8955:
1085:. Here the definition of homotopy relies on a notion of continuity, and hence a topology.
8:
20542:
20151:
20077:
18245:
17931:
15861:
15384:
14795:
14769:
14272:{\displaystyle \{w_{i_{1},\dots ,i_{k}}\,\vert \,1\leq i_{1}<\cdots <i_{k}\leq n\}}
12253:
12110:
7635:
6775:
5381:
153:
20960:
20881:
20617:
20429:
20390:
18633:
18606:
15281:, but they are not algebraically independent. They are equivalent to the statement that
10938:
8118:
The quickest way of giving the Grassmannian a geometric structure is to express it as a
21065:
21020:
20972:
20946:
20893:
20867:
20840:
20822:
20795:
20720:
20558:
20523:
20488:
20480:
20345:
20315:
20145:
20037:
may be calculated in the planar limit via a positive Grassmannian construct called the
20030:
19981:
19820:
19669:
19649:
19629:
19327:
18684:
18664:
18489:
18406:
18386:
18362:
18342:
18319:
18309:
18287:
18227:
18204:
18184:
18000:
17976:
17694:
17509:
16606:
16559:
16424:
16187:
16167:
15889:
15780:
15278:
14556:
14536:
14434:
14385:
14012:
13758:
13686:
13529:
13509:
13463:
13391:
13371:
13143:
13140:
is well-defined. To see that it is an embedding, notice that it is possible to recover
13095:
12945:
12844:
12824:
12804:
12670:
12617:
12180:
12136:
12037:
11964:
11518:
11097:
11077:
10918:
10858:
10834:
10790:
10770:
10678:
10600:
10580:
10478:
10458:
10438:
10418:
10368:
9886:
9739:
9467:
9230:
9170:
9112:
9092:
8965:
8892:
8780:
8392:
8238:
8185:
8165:
7920:
7801:
7735:
7676:
7617:
7407:
7127:
7029:
6875:
6816:
6757:
6527:
6495:
5794:
5774:
5663:
5643:
5482:
5413:
5324:
5304:
5284:
5264:
4912:
3760:
3447:
3141:
3094:
2995:
2814:
2786:
2577:
2557:
2537:
2517:
2461:
2208:
2188:
2120:
2096:
2021:
2001:
1955:
1734:
1649:
1629:
1470:
1383:
1284:
1264:
1074:
1052:
1032:
1001:
981:
881:
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720:
700:
641:
621:
601:
293:
273:
219:
158:
135:
112:
85:
20314:
Witten, Edward (1993). "The Verlinde algebra and the cohomology of the Grassmannian".
18284:-dimensional orthogonal complements of these planes yield an orthogonal vector bundle
16151:{\displaystyle 0\rightarrow (V/W)^{*}\rightarrow V^{*}\rightarrow W^{*}\rightarrow 0.}
14111:
13064:
11330:
Since the Grassmannian scheme represents a functor, it comes with a universal object,
10330:{\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{n})=U(n)/\left(U(k)\times U(n-k)\right).}
9867:{\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{n})=O(n)/\left(O(k)\times O(n-k)\right).}
3467:
2041:
21307:
21271:
21249:
21239:
21210:
21191:
21169:
21109:
21093:
21049:
20994:
20836:
20799:
20785:
20724:
20710:
20673:
20629:
20492:
20349:
20279:
20263:
20230:
20125:
20071:
20061:
19985:
19977:
19958:
17404:
16842:
16672:
9067:
it also becomes possible to use smaller groups in this construction. To do this over
8951:
8159:
8119:
5728:
2114:
1924:
1731:. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices
430:
75:
21069:
20897:
20844:
10915:. By construction, the Grassmannian scheme is compatible with base changes: for any
6550:
368:
21299:
21285:
21127:
21041:
20976:
20964:
20885:
20832:
20777:
20751:
20702:
20663:
20621:
20568:
20515:
20470:
20462:
20433:
20394:
20297:
20156:
20097:
18997:{\displaystyle \gamma _{k,n}(A)=\theta _{n}\{g\in \operatorname {O} (n):gw\in A\}.}
18774:
18249:
17400:
9130:
7694:
6834:
4168:
between any two such coordinate neighborhoods, the affine coordinate matrix values
1620:
1195:
1116:
237:
20573:
20546:
21265:
21185:
21119:
21057:
20359:
20339:
20289:
20226:
20161:
20141:
20045:
13117:
10496:
8944:
8883:
6318:
2808:
2804:
694:
322:
106:
20968:
20506:
Narasimhan, M. S.; Ramanan, S. (1963). "Existence of Universal Connections II".
13327:
embeds as a nonsingular projective algebraic subvariety of the projectivization
21261:
20668:
20651:
9394:{\displaystyle O(w_{0},q|_{w_{0}})\times O(w_{0}^{\perp },q|_{w_{0}^{\perp }})}
8983:
3933:{\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}:=W(W_{i_{1},\dots ,i_{k}})^{-1}}
3701:
690:
315:
21303:
20889:
20475:
19944:
16343:{\displaystyle \mathbf {Gr} _{k}(V)\leftrightarrow \mathbf {Gr} {(n-k},V^{*})}
15474:{\displaystyle \iota (\mathbf {Gr} _{2}(V)\subset \mathbf {P} (\Lambda ^{2}V)}
5727:
as its image. Since the rank of an orthogonal projection operator equals its
21324:
21253:
21139:
21053:
20677:
20633:
20453:
Narasimhan, M. S.; Ramanan, S. (1961). "Existence of Universal Connections".
20091:
20038:
19310:
18493:
18222:
17044:
13280:
of the Grassmannian satisfies a set of simple quadratic relations called the
11291:
10341:
9904:
6521:
6315:
6307:{\displaystyle \{X\in \mathrm {End} (V)\mid \mathrm {tr} (XX^{\dagger })=k\}}
5227:
1171:
1026:
793:
319:
13120:
matrix). Since the right-hand side takes values in the projectivized space,
5018:. The transition functions are therefore rational in the matrix elements of
21161:
19993:
19989:
18770:
16668:
11580:
8887:
8156:
6362:
2182:
130:
20781:
20706:
21223:
21157:
20769:
20694:
20625:
19954:
19943:
with respect to a real or complex scalar product are closely related to
18428:
18313:
13190:
13113:
6032:
this gives completely explicit equations for embedding the Grassmannians
667:
311:
27:
18630:
which violates the degree of the cohomology corresponding to a state by
18221:-space. Fibering these planes over the Grassmannian one arrives at the
17410:
As an example of the technique, consider the problem of determining the
15960:{\displaystyle 0\rightarrow W\rightarrow V\rightarrow V/W\rightarrow 0.}
8774:
21045:
20755:
20527:
20484:
20320:
20018:
Finite dimensional positive Grassmann manifolds can be used to express
18305:
15973:
11900:
10852:
10565:{\displaystyle {\mathcal {E}}_{T}:={\mathcal {E}}\otimes _{O_{S}}O_{T}}
1082:
952:. (In order to do this, we have to translate the tangent space at each
679:
19810:{\displaystyle \mathbf {Gr} _{k}^{0}(V,Q)\subset \mathbf {Gr} _{k}(V)}
16667:
The detailed study of Grassmannians makes use of a decomposition into
12991:
be the projectivization of the wedge product of these basis elements:
5213:
20438:
20413:
20398:
18627:
18600:
8879:
6229:
Since this defines the Grassmannian as a closed subset of the sphere
5226:
problem 5-C). For this, choose a positive definite real or Hermitian
1078:
1019:
20913:"A Mathematician's Unanticipated Journey Through the Physical World"
20739:
20519:
20466:
20185:
16263:. In terms of the Grassmannian, this gives a canonical isomorphism
12763:\iota :\mathbf {Gr} (k,V)\to \mathbf {P} \left(\Lambda ^{k}V\right).
4876:{\displaystyle {\hat {A}}_{j_{1},\dots ,j_{k}}^{i_{1},\dots ,i_{k}}}
3945:
the homogeneous coordinate matrix having the identity matrix as the
20547:"Schubert Calculus and representations of the general linear group"
20129:
14451:
under the Plücker map, relative to the basis of the exterior power
11283:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}}\otimes _{O_{S}}K(s))}
675:
20951:
20872:
20827:
20563:
19475:{\displaystyle {\widetilde {\mathbf {Gr} }}_{k}(\mathbf {R} ^{n})}
2807:. We may apply column operations to reduce this submatrix to the
1069:
to a suitably generalised Grassmannian—although various embedding
682:
collections of subspaces. Giving them the further structure of a
20937:
Arkani-Hamed, Nima; Trnka, Jaroslav (2013). "The Amplituhedron".
20019:
18701:-dimensional Euclidean space, we may define a uniform measure on
18424:
10672:
7890:
are orthonomal. The formula for the orthogonal projection matrix
5162:{\displaystyle \{U_{i_{1},\dots ,i_{k}},A^{i_{1},\dots ,i_{k}}\}}
4161:{\displaystyle U_{i_{1},\dots ,i_{k}}\cap U_{j_{1},\dots ,j_{k}}}
1070:
20656:
Publications of the Research Institute for Mathematical Sciences
17033:{\displaystyle \lambda _{1}\geq \cdots \geq \lambda _{k}\geq 0,}
16784:{\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V}
15310:
is the projectivization of a completely decomposable element of
11508:{\displaystyle {\mathcal {E}}_{\mathbf {Gr} (k,{\mathcal {E}})}}
1211:
20662:(3). European Mathematical Society Publishing House: 943–1001.
19961:, is cut out as the intersection of a number of quadrics, the
19948:
15678:
In general, many more equations are needed to define the image
15191:{\displaystyle j_{1},\ldots ,{\widehat {j_{l}}},\ldots j_{k+1}}
5210:
as a differentiable manifold and also as an algebraic variety.
1018:-dimensional vector subspace. This idea is very similar to the
19626:
Given a real or complex nondegenerate symmetric bilinear form
18126:
10212:{\displaystyle w_{0}=\mathbf {C} ^{k}\subset \mathbf {C} ^{n}}
9729:{\displaystyle w_{0}=\mathbf {R} ^{k}\subset \mathbf {R} ^{n}}
7427:
An analogous construction applies to the complex Grassmannian
1049:, so that every vector bundle generates a continuous map from
436:
Notations for Grassmannians vary between authors, and include
426:
19686:(i.e., a scalar product), the totally isotropic Grassmannian
17148:{\displaystyle X_{\lambda }(k,n)\subset \mathbf {Gr} _{k}(V)}
20173:
16898:{\displaystyle \lambda =(\lambda _{1},\cdots ,\lambda _{k})}
14759:{\displaystyle 1\leq j_{1}<j_{2}\cdots <j_{k+1}\leq n}
14685:{\displaystyle 1\leq i_{1}<i_{2}\cdots <i_{k-1}\leq n}
13271:
8101:{\displaystyle P_{w}:=\sum _{i=1}^{k}w_{i}w_{i}^{\dagger }.}
7535:{\displaystyle P(k,n,\mathbf {C} )\subset M(n,\mathbf {C} )}
7173:
spanned by its columns and, conversely, sending any element
6701:{\displaystyle P(k,n,\mathbf {R} )\subset M(n,\mathbf {R} )}
1201:
The simplest Grassmannian that is not a projective space is
16468:
therefore determines a (non-canonical) isomorphism between
15572:{\displaystyle (w_{12},w_{13},w_{14},w_{23},w_{24},w_{34})}
12250:
correspond to the projective linear subspaces of dimension
5621:
3537:
does not affect the values of the affine coordinate matrix
2992:
for which, for any choice of homogeneous coordinate matrix
8409:
is determined only up to right multiplication by elements
8113:
1257:
with the structure of a differentiable manifold, choose a
367:
The earliest work on a non-trivial Grassmannian is due to
19009:
This measure is invariant under the action of the group
15666:{\displaystyle w_{12}w_{34}-w_{13}w_{24}+w_{14}w_{23}=0.}
15277:
omitted. These are consistent, determining a nonsingular
12415:\mathbf {P} (V)\times _{K}\mathbf {Gr} (k,{\mathcal {E}})
12098:{\displaystyle \mathbf {P} ({\mathcal {E}})\times _{S}T.}
9484:. This gives an identification as the homogeneous space
8947:, and this construction shows that the Grassmannian is a
3418:, written in the same order. The choice of homogeneous
17399:
These are affine spaces, and their closures (within the
2225:
are linearly independent, the result will have the form
816:
can be considered as a subspace of the tangent space of
666:
By giving a collection of subspaces of a vector space a
21207:
Differential geometry, Lie groups, and symmetric spaces
17684:{\displaystyle \mathbf {Gr} _{k-1}(\mathbf {R} ^{n-1})}
11583:
induces a closed immersion from the projective bundle:
360:. In general they have the structure of a nonsingular
20603:
20540:
18057:
16982:
consisting of weakly decreasing non-negative integers
16970:{\displaystyle |\lambda |=\sum _{i=1}^{k}\lambda _{i}}
13838:
7221:{\displaystyle w\in \mathbf {Gr} (k,\mathbf {R} ^{n})}
6321:
space. This construction also turns the Grassmannian
4076:
in the consecutive complementary rows. On the overlap
2242:
20699:
Tau functions and Their Applications, Chapts. 4 and 5
19874:
19843:
19823:
19744:
19692:
19672:
19652:
19632:
19621:
19494:
19429:
19379:
19350:
19330:
19282:
19269:{\displaystyle \gamma _{k,n}(\mathbf {Gr} _{k}(V))=1}
19213:
19162:
19127:
19049:
19015:
18910:
18858:
18811:
18782:
18748:
18707:
18687:
18667:
18636:
18609:
18507:
18442:
18409:
18389:
18365:
18345:
18322:
18290:
18258:
18230:
18207:
18187:
18137:
18032:
18003:
17979:
17940:
17778:
17760:{\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n-1})}
17717:
17697:
17635:
17569:
17532:
17512:
17452:
17419:
17250:
17212:
17161:
17092:
17053:
16991:
16919:
16853:
16797:
16732:
16687:
16629:
16609:
16582:
16562:
16515:
16474:
16447:
16427:
16387:
16361:
16274:
16242:
16210:
16190:
16170:
16085:
16054:
16027:
15986:
15921:
15892:
15864:
15829:
15803:
15783:
15734:
15684:
15590:
15487:
15413:
15387:
15349:
15316:
15287:
15256:
15204:
15127:
14928:
14882:
14832:
14798:
14772:
14698:
14624:
14580:
14559:
14539:
14487:
14457:
14437:
14408:
14388:
14336:
14289:
14180:
14114:
14088:
14035:
14015:
14002:{\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}
13957:
13786:
13761:
13709:
13689:
13662:
13604:
13552:
13532:
13512:
13486:
13466:
13414:
13394:
13374:
13333:
13292:
13230:
13199:
13166:
13146:
13126:
13098:
13067:
12997:
12968:
12948:
12896:
12867:
12847:
12827:
12807:
12781:
12673:
12643:
12620:
12580:
12438:
12313:
12282:
12256:
12203:
12183:
12159:
12139:
12113:
12060:
12040:
11987:
11967:
11936:
11909:
11821:
11798:{\displaystyle T\to \mathbf {Gr} (k,{\mathcal {E}}),}
11758:
11594:
11541:
11521:
11466:
11442:
11360:
11336:
11299:
11223:
11172:
11120:
11100:
11080:
10971:
10941:
10921:
10881:
10861:
10851:
and we recover the usual Grassmannian variety of the
10837:
10813:
10793:
10773:
10749:
10701:
10681:
10623:
10603:
10583:
10510:
10481:
10461:
10441:
10421:
10393:
10228:
10168:
10133:
9950:
9912:
9889:
9765:
9742:
9685:
9650:
9493:
9470:
9443:
9411:
9289:
9253:
9233:
9193:
9173:
9138:
9115:
9095:
9073:
9051:
9029:
8992:
8968:
8915:
8895:
8886:
under the quotient structure. More generally, over a
8850:
8828:
8806:
8783:
8763:{\displaystyle \mathbf {Gr} (k,V)=\mathrm {GL} (V)/H}
8708:
8665:
8628:
8481:
8451:
8415:
8395:
8355:
8310:
8261:
8241:
8208:
8188:
8168:
8128:
8035:
7978:
7943:
7923:
7896:
7844:
7810:
7758:
7738:
7703:
7679:
7644:
7620:
7574:
7548:
7482:
7433:
7410:
7375:
7323:
7305:{\displaystyle P_{w}:=\sum _{i=1}^{k}w_{i}w_{i}^{T},}
7239:
7179:
7150:
7130:
7081:
7052:
7032:
6983:
6940:
6898:
6878:
6843:
6819:
6784:
6760:
6714:
6648:
6622:
6585:
6530:
6498:
6461:
6376:
6327:
6235:
6197:
6162:
6136:
6087:
6038:
6003:
5968:
5822:
5797:
5777:
5737:
5686:
5666:
5646:
5537:
5505:
5485:
5441:
5416:
5390:
5347:
5327:
5307:
5287:
5267:
5235:
5175:
5077:
5024:
4962:
4935:
4915:
4889:
4796:
4586:
4528:
4475:
4285:
4227:
4174:
4082:
4029:
3977:
3951:
3826:
3783:
3763:
3710:
3653:
3600:
3543:
3496:
3470:
3450:
3424:
3372:
3308:
3255:
3217:
3164:
3144:
3117:
3097:
3044:
3018:
2998:
2951:
2898:
2885:{\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}
2840:
2817:
2789:
2737:
2684:
2658:
2645:{\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}
2600:
2580:
2560:
2540:
2520:
2484:
2464:
2426:
2236:
2211:
2191:
2143:
2123:
2099:
2073:
2044:
2024:
2004:
1978:
1958:
1932:
1881:
1838:
1786:
1757:
1737:
1699:
1672:
1652:
1632:
1599:
1552:
1496:
1473:
1432:
1406:
1386:
1334:
1307:
1287:
1267:
1222:
1124:
1055:
1035:
1004:
984:
958:
908:
884:
851:
822:
802:
772:
743:
723:
703:
644:
624:
604:
564:
523:
483:
442:
377:
331:
296:
276:
245:
222:
181:
161:
138:
115:
88:
39:
21234:. Vol. 218 (Second ed.). New York London:
20100:
are bundles of orthonormal frames over Grassmanians.
20044:
Grassmann manifolds have also found applications in
17556:{\displaystyle \mathbf {R} \subset \mathbf {R} ^{n}}
11159:{\displaystyle \{s\}={\text{Spec}}K(s)\rightarrow S}
7972:spanned by the orthonormal (unitary) basis vectors
6443:{\displaystyle d(w,w'):=\lVert P_{w}-P_{w'}\rVert ,}
4575:
are invertible. This may equivalently be written as
3158:
is nonsingular. The affine coordinate functions on
21104:. Wiley Classics Library (2nd ed.). New York:
20989:Pavan Turaga, Ashok Veeraraghavan, Rama Chellappa:
20274:, Wiley Classics Library (2nd ed.), New York:
19416:{\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}
18174:{\displaystyle \mathbf {Gr} _{k}(\mathbf {C} ^{n})}
17606:{\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}
17489:{\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}
11899:Conversely, any such closed immersion comes from a
11210:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})_{s}}
10661:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}}_{T})}
10340:In particular, this shows that the Grassmannian is
5341:determines a unique orthogonal projection operator
5214:
The Grassmannian as a set of orthogonal projections
945:{\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}
414:{\displaystyle \mathbf {Gr} _{2}(\mathbf {R} ^{4})}
19927:
19855:
19829:
19809:
19730:
19678:
19658:
19638:
19610:
19474:
19415:
19365:
19336:
19301:
19268:
19199:
19148:
19109:{\displaystyle \gamma _{k,n}(gA)=\gamma _{k,n}(A)}
19108:
19030:
18996:
18891:
18844:
18797:
18761:
18734:
18693:
18673:
18645:
18618:
18582:
18475:
18415:
18395:
18371:
18351:
18328:
18296:
18276:
18236:
18213:
18193:
18173:
18110:
18009:
17985:
17965:
17917:
17759:
17703:
17683:
17605:
17555:
17518:
17488:
17438:
17386:
17231:
17194:
17147:
17078:
17032:
16969:
16897:
16833:
16783:
16714:
16647:
16615:
16595:
16568:
16548:
16501:
16460:
16433:
16413:
16373:
16342:
16255:
16228:
16196:
16176:
16150:
16067:
16040:
16013:
15959:
15898:
15878:
15847:
15815:
15789:
15761:
15720:
15665:
15571:
15473:
15399:
15373:
15332:
15302:
15269:
15242:
15190:
15108:
14909:
14868:
14810:
14784:
14758:
14684:
14607:
14565:
14545:
14525:
14473:
14443:
14423:
14394:
14374:
14322:
14271:
14166:
14100:
14074:
14021:
14001:
13938:
13767:
13747:
13695:
13675:
13648:
13590:
13538:
13518:
13498:
13472:
13452:
13400:
13380:
13360:
13319:
13257:
13211:
13181:
13152:
13132:
13104:
13081:
13053:
12983:
12954:
12934:
12882:
12853:
12833:
12813:
12793:
12762:
12679:
12659:
12626:
12606:
12546:
12414:
12343:
12299:
12268:
12243:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})(K)}
12242:
12189:
12169:
12145:
12125:
12097:
12046:
12027:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})(T)}
12026:
11973:
11953:
11922:
11886:
11797:
11730:
11571:
11527:
11507:
11452:
11428:
11346:
11314:
11282:
11209:
11158:
11106:
11086:
11061:
10952:
10927:
10907:
10867:
10843:
10823:
10799:
10779:
10759:
10731:
10687:
10660:
10609:
10589:
10564:
10487:
10467:
10447:
10427:
10403:
10329:
10211:
10154:
10116:
9933:
9895:
9866:
9748:
9728:
9671:
9632:
9476:
9456:
9429:
9393:
9272:
9239:
9219:
9179:
9159:
9121:
9101:
9081:
9059:
9037:
9012:
8974:
8935:
8901:
8870:
8836:
8814:
8789:
8762:
8691:
8648:
8611:
8464:
8433:
8401:
8381:
8338:
8293:
8247:
8227:
8194:
8174:
8148:
8100:
8016:
7964:
7929:
7909:
7882:
7830:
7788:
7744:
7722:
7685:
7663:
7626:
7603:
7560:
7534:
7468:
7416:
7396:
7361:
7304:
7220:
7165:
7136:
7116:
7067:
7038:
7018:
6969:
6922:
6884:
6862:
6825:
6803:
6766:
6743:
6700:
6634:
6608:
6536:
6504:
6484:
6442:
6353:
6314:this is one way to see that the Grassmannian is a
6306:
6218:
6183:
6148:
6122:
6073:
6024:
5989:
5949:
5803:
5783:
5763:
5719:
5672:
5652:
5627:
5518:
5491:
5466:
5422:
5402:
5372:
5333:
5313:
5293:
5273:
5253:
5202:
5161:
5063:
5010:
4948:
4921:
4901:
4875:
4777:
4567:
4514:
4456:
4266:
4213:
4160:
4068:
4015:
3963:
3932:
3807:
3769:
3749:
3692:
3639:
3582:
3529:
3482:
3456:
3436:
3410:
3358:
3294:
3241:
3203:
3150:
3130:
3103:
3083:
3030:
3004:
2984:
2937:
2884:
2823:
2795:
2775:
2723:
2670:
2644:
2586:
2566:
2546:
2526:
2506:
2470:
2450:
2407:
2217:
2197:
2173:
2129:
2105:
2085:
2056:
2030:
2010:
1990:
1964:
1944:
1911:
1862:
1819:
1772:
1743:
1723:
1685:
1658:
1638:
1611:
1585:
1534:
1479:
1459:
1418:
1392:
1372:
1320:
1293:
1273:
1249:
1145:
1061:
1041:
1010:
990:
970:
944:
890:
866:
837:
808:
784:
758:
729:
709:
650:
630:
610:
590:
550:
509:
469:
413:
352:
302:
282:
262:
228:
208:
167:
144:
121:
94:
66:
21267:Geometry of Sets and Measures in Euclidean Spaces
20936:
20812:
20505:
20452:
14402:). These are the linear coordinates of the image
11810:this closed immersion induces a closed immersion
10495:, the Grassmannian functor associates the set of
7831:{\displaystyle \langle \,\cdot ,\cdot \,\rangle }
7469:{\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{n})}
7019:{\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{n})}
6123:{\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{N})}
6074:{\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{N})}
1174:to that plane (and vice versa); hence the spaces
978:so that it passes through the origin rather than
686:, one can talk about smooth choices of subspace.
21322:
20652:"Solitons and infinite-dimensional Lie algebras"
20225:. Vol. 1 (2nd ed.). Berlin, New York:
2038:, for which the equivalence classes are denoted
427:§ Plücker coordinates and Plücker relations
21092:
20424:(9). American Institute of Physics: 1945–1970.
20385:(9). American Institute of Physics: 3197–3208.
20262:
19485:As a homogeneous space it can be expressed as:
14330:of the Grassmannian (with respect to the basis
12344:{\displaystyle \mathbf {P} ({\mathcal {G}})(K)}
11572:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})}
10732:{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})}
10362:
8122:. First, recall that the general linear group
7804:is with respect to the Hermitian inner product
5011:{\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}}
2811:, and the remaining entries uniquely determine
1972:. This defines an equivalence relation between
21156:
21018:
20857:
20774:Tau functions and Their Applications, Chapt. 7
20203:
20066:For an example of the use of Grassmannians in
19928:{\displaystyle Q(u,v)=0,\,\forall \,u,v\in w.}
18595:reflecting the existence in the corresponding
18131:Every point in the complex Grassmann manifold
12034:are exactly the projective subbundles of rank
8882:, this construction makes the Grassmannian a
5223:
20557:(4). American Mathematical Society: 909–940.
20411:
20376:
19957:variety which, similarly to the image of the
18892:{\displaystyle A\subset \mathbf {Gr} _{k}(V)}
18583:{\displaystyle c_{k}(E)c_{n-k}(F)=(-1)^{n-k}}
7476:, identifying it bijectively with the subset
5254:{\displaystyle \langle \cdot ,\cdot \rangle }
3700:may take arbitrary values, and they define a
1212:The Grassmannian as a differentiable manifold
1025:This can with some effort be extended to all
421:, parameterizing them by what are now called
20768:
20693:
20612:(11). Physical Society of Japan: 3806–3812.
20215:
20144:, the Grassmannian plays a similar role for
18988:
18949:
17378:
17279:
17226:
17213:
15721:{\displaystyle \iota (\mathbf {Gr} _{k}(V))}
14869:{\displaystyle \iota (\mathbf {Gr} _{k}(V))}
14266:
14221:
14181:
11127:
11121:
8586:
8521:
8428:
8416:
7825:
7811:
6934:There is a bijective correspondence between
6434:
6403:
6301:
6236:
5248:
5236:
5156:
5078:
3359:{\displaystyle WW_{i_{1},\dots ,i_{k}}^{-1}}
1111:is the space of lines through the origin in
216:is the space of lines through the origin in
21284:
20191:
18379:. Then the relations merely state that the
18127:Cohomology ring of the complex Grassmannian
15976:to each of these three spaces and the dual
15762:{\displaystyle \mathbf {P} (\Lambda ^{k}V)}
14608:{\displaystyle \mathbf {P} (\Lambda ^{k}V)}
13361:{\displaystyle \mathbf {P} (\Lambda ^{k}V)}
12574:is a natural embedding of the Grassmannian
10371:, the Grassmannian can be constructed as a
9941:acts transitively, and we find analogously
2831:. Hence we have the following definition:
2594:, there exists an ordered set of integers
2067:We now define a coordinate atlas. For any
19731:{\displaystyle \mathbf {Gr} _{k}^{0}(V,Q)}
13089:denotes the projective equivalence class.
9883:, if we choose an Hermitian inner product
1380:, viewed as column vectors. Then for any
670:structure, it is possible to talk about a
20950:
20871:
20826:
20667:
20649:
20572:
20562:
20474:
20437:
20319:
19906:
19902:
18845:{\displaystyle w\in \mathbf {Gr} _{k}(V)}
17318:
17312:
17195:{\displaystyle W\in \mathbf {Gr} _{k}(V)}
14323:{\displaystyle w\in \mathbf {Gr} _{k}(V)}
14224:
14220:
13813:
13272:Plücker coordinates and Plücker relations
13075:
13071:
8553:
8547:
7965:{\displaystyle w\subset \mathbf {C} ^{n}}
7824:
7814:
7397:{\displaystyle w\subset \mathbf {R} ^{n}}
5915:
5731:, we can identify the Grassmann manifold
4274:are related by the transition relations
3530:{\displaystyle w\in \mathbf {Gr} _{k}(V)}
2985:{\displaystyle w\in \mathbf {Gr} _{k}(V)}
1820:{\displaystyle w\in \mathbf {Gr} _{k}(V)}
1586:{\displaystyle w\in \mathbf {Gr} _{k}(V)}
1198:) may all be identified with each other.
433:later introduced the concept in general.
21270:. New York: Cambridge University Press.
21204:
20910:
20645:
20643:
20606:Journal of the Physical Society of Japan
20599:
20597:
19316:
18308:of the Grassmannians is generated, as a
12559:
10382:
8962:as an algebraic variety. In particular,
7117:{\displaystyle P\in P(k,n,\mathbf {R} )}
6923:{\displaystyle \operatorname {tr} (P)=k}
6219:{\displaystyle \mathbf {C} ^{n\times n}}
6184:{\displaystyle \mathbf {R} ^{n\times n}}
3211:are then defined as the entries of the
2554:rows need not be independent, but since
1022:for surfaces in a 3-dimensional space.)
21260:
21138:
19321:This is the manifold consisting of all
16834:{\displaystyle \mathrm {dim} (V_{i})=i}
13649:{\displaystyle (w_{i1},\cdots ,w_{in})}
8954:. It follows from the existence of the
8294:{\displaystyle w\in \mathbf {Gr} (k,V)}
8114:The Grassmannian as a homogeneous space
5720:{\displaystyle w_{P}:=\mathrm {Im} (P)}
21323:
21183:
20689:
20687:
20337:
20313:
16549:{\displaystyle \mathbf {Gr} _{n-k}(V)}
14075:{\displaystyle w_{i_{1},\dots ,i_{k}}}
11166:induces an isomorphism from the fiber
8202:. Therefore, if we choose a subspace
7604:{\displaystyle P\in M(n,\mathbf {C} )}
6744:{\displaystyle P\in M(n,\mathbf {R} )}
5640:Conversely, every projection operator
5064:{\displaystyle A^{i_{1},\dots ,i_{k}}}
4568:{\displaystyle W_{j_{1},\dots ,j_{k}}}
4515:{\displaystyle W_{i_{1},\dots ,i_{k}}}
4267:{\displaystyle A^{j_{1},\dots ,j_{k}}}
4214:{\displaystyle A^{i_{1},\dots ,i_{k}}}
4069:{\displaystyle A^{i_{1},\dots ,i_{k}}}
3750:{\displaystyle U_{i_{1},\dots ,i_{k}}}
3693:{\displaystyle A^{i_{1},\dots ,i_{k}}}
3640:{\displaystyle U_{i_{1},\dots ,i_{k}}}
3583:{\displaystyle A^{i_{1},\dots ,i_{k}}}
3295:{\displaystyle A^{i_{1},\dots ,i_{k}}}
3204:{\displaystyle U_{i_{1},\dots ,i_{k}}}
3084:{\displaystyle W_{i_{1},\dots ,i_{k}}}
2938:{\displaystyle U_{i_{1},\dots ,i_{k}}}
2724:{\displaystyle W_{i_{1},\dots ,i_{k}}}
1467:, we may choose a basis consisting of
20650:Jimbo, Michio; Miwa, Tetsuji (1983).
20640:
20594:
18656:
18431:allows one to write this relation as
17629:and those that do not. The former is
16722:are defined in terms of a specified
15243:{\displaystyle j_{1},\ldots ,j_{k+1}}
14526:{\displaystyle (e_{1},\cdots ,e_{n})}
14375:{\displaystyle (e_{1},\cdots ,e_{n})}
13591:{\displaystyle (w_{1},\cdots ,w_{k})}
13453:{\displaystyle (e_{1},\cdots ,e_{n})}
12935:{\displaystyle (w_{1},\cdots ,w_{k})}
10344:, and of (real or complex) dimension
8382:{\displaystyle g\in \mathrm {GL} (V)}
8017:{\displaystyle (w_{1},\cdots ,w_{k})}
7883:{\displaystyle (e_{1},\cdots ,e_{n})}
7362:{\displaystyle (w_{1},\cdots ,w_{k})}
3647:. Moreover, the coordinate matrices
2776:{\displaystyle (i_{1},\ldots ,i_{k})}
898:its tangent space defines a map from
21080:, see section 4.3., pp. 137–140
21033:Publications Mathématiques de l'IHÉS
20911:Hartnett, Kevin (16 December 2020).
20737:
20581:
20333:
20331:
18735:{\displaystyle \mathbf {Gr} _{k}(V)}
16715:{\displaystyle \mathbf {Gr} _{k}(V)}
16502:{\displaystyle \mathbf {Gr} _{k}(V)}
14910:{\displaystyle \mathbf {Gr} _{k}(V)}
14826:, are valid and determine the image
13748:{\displaystyle (W^{1},\dots ,W^{n})}
13683:with respect to the chosen basis of
13320:{\displaystyle \mathbf {Gr} _{k}(V)}
13286:. These show that the Grassmannian
7838:in which the standard basis vectors
5467:{\displaystyle V=w\oplus w^{\perp }}
5373:{\displaystyle P_{w}:V\rightarrow V}
5203:{\displaystyle \mathbf {Gr} _{k}(V)}
4016:{\displaystyle (i_{1},\dots ,i_{k})}
3411:{\displaystyle (i_{1},\dots ,i_{k})}
1535:{\displaystyle (W_{1},\dots ,W_{k})}
1460:{\displaystyle \mathbf {Gr} _{k}(V)}
1373:{\displaystyle (e_{1},\dots ,e_{n})}
1281:. This is equivalent to identifying
1250:{\displaystyle \mathbf {Gr} _{k}(V)}
551:{\displaystyle \mathbf {Gr} _{k}(n)}
470:{\displaystyle \mathbf {Gr} _{k}(V)}
209:{\displaystyle \mathbf {Gr} _{1}(V)}
67:{\displaystyle \mathbf {Gr} _{k}(V)}
21222:
20762:
20684:
20534:
20405:
20179:
19200:{\displaystyle \theta _{n}(O(n))=1}
13258:{\displaystyle v\wedge \iota (w)=0}
11325:
9756:components) we get the isomorphism
6970:{\displaystyle P(k,n,\mathbf {R} )}
6751:that satisfy the three conditions:
3302:whose rows are those of the matrix
13:
21187:Algebraic Geometry: A First Course
20370:
19903:
19622:Orthogonal isotropic Grassmannians
18958:
16805:
16802:
16799:
16414:{\displaystyle W^{0}\subset V^{*}}
15744:
15579:, this single Plücker relation is
15456:
15318:
14590:
14459:
13343:
12741:
12645:
12607:{\displaystyle \mathbf {Gr} (k,V)}
12507:
12405:
12324:
12223:
12162:
12071:
12007:
11954:{\displaystyle {\mathcal {E}}_{T}}
11940:
11860:
11833:
11784:
11717:
11680:
11653:
11628:
11605:
11561:
11495:
11470:
11445:
11408:
11383:
11339:
11243:
11192:
11040:
10991:
10960:, we have a canonical isomorphism
10908:{\displaystyle \mathbf {Gr} (k,V)}
10816:
10752:
10721:
10644:
10530:
10514:
10396:
10155:{\displaystyle V=\mathbf {C} ^{n}}
9672:{\displaystyle V=\mathbf {R} ^{n}}
9220:{\displaystyle \mathbf {Gr} (k,V)}
8997:
8994:
8920:
8917:
8855:
8852:
8739:
8736:
8692:{\displaystyle \mathbf {Gr} (k,V)}
8633:
8630:
8596:
8593:
8534:
8531:
8498:
8495:
8492:
8489:
8366:
8363:
8133:
8130:
7781:
7772:
7765:
7762:
6544:, and hence an equivalent metric.
6354:{\displaystyle \mathbf {Gr} (k,V)}
6272:
6269:
6252:
6249:
6246:
6025:{\displaystyle V=\mathbf {C} ^{n}}
5990:{\displaystyle V=\mathbf {R} ^{n}}
5920:
5917:
5867:
5864:
5861:
5764:{\displaystyle \mathbf {Gr} (k,V)}
5704:
5701:
1328:, with the standard basis denoted
1146:{\displaystyle \mathbf {P} ^{n-1}}
693:of smooth manifolds embedded in a
591:{\displaystyle \mathbf {Gr} (k,n)}
510:{\displaystyle \mathbf {Gr} (k,V)}
175:. For example, the Grassmannian
82:that parameterizes the set of all
14:
21357:
20328:
20307:
20256:
19976:Another important application is
17767:. This gives recursive formulae:
16662:
16355:that associates to each subspace
14917:under the Plücker map embedding:
12614:into the projectivization of the
11961:to a locally free module of rank
7789:{\displaystyle {\rm {{tr}(P)=k}}}
6609:{\displaystyle M(n,\mathbf {R} )}
4023:and the affine coordinate matrix
2834:For each ordered set of integers
1115:-space, so it is the same as the
1088:
21228:Introduction to Smooth Manifolds
21102:Principles of algebraic geometry
20837:10.1111/j.1467-9590.2009.00448.x
20412:Harnad, J.; Shnider, S. (1995).
20272:Principles of algebraic geometry
20222:Éléments de géométrie algébrique
19788:
19785:
19750:
19747:
19698:
19695:
19524:
19503:
19500:
19459:
19438:
19435:
19400:
19385:
19382:
19366:{\displaystyle \mathbf {R} ^{n}}
19353:
19238:
19235:
18870:
18867:
18823:
18820:
18713:
18710:
18158:
18143:
18140:
17738:
17723:
17720:
17662:
17641:
17638:
17590:
17575:
17572:
17543:
17534:
17473:
17458:
17455:
17293:
17290:
17173:
17170:
17126:
17123:
17047:fits within the rectangular one
16693:
16690:
16521:
16518:
16480:
16477:
16306:
16303:
16280:
16277:
15736:
15696:
15693:
15448:
15425:
15422:
14888:
14885:
14844:
14841:
14582:
14301:
14298:
13335:
13298:
13295:
12731:
12708:
12705:
12585:
12582:
12492:
12489:
12463:
12390:
12387:
12363:
12315:
12284:
12208:
12205:
12107:Under this identification, when
12062:
11992:
11989:
11851:
11823:
11769:
11766:
11702:
11699:
11671:
11638:
11635:
11617:
11596:
11546:
11543:
11480:
11477:
11436:and therefore a quotient module
11393:
11390:
11365:
11362:
11228:
11225:
11177:
11174:
11024:
11021:
10976:
10973:
10886:
10883:
10706:
10703:
10675:is representable by a separated
10628:
10625:
10248:
10233:
10230:
10199:
10184:
10142:
9955:
9952:
9785:
9770:
9767:
9716:
9701:
9659:
9498:
9495:
9437:is the orthogonal complement of
9198:
9195:
9167:acts transitively on the set of
9089:, fix a Euclidean inner product
9075:
9053:
9031:
9013:{\displaystyle \mathrm {GL} (V)}
8936:{\displaystyle \mathrm {GL} (V)}
8871:{\displaystyle \mathrm {GL} (V)}
8830:
8808:
8713:
8710:
8670:
8667:
8649:{\displaystyle \mathrm {GL} (V)}
8272:
8269:
8149:{\displaystyle \mathrm {GL} (V)}
7952:
7594:
7525:
7502:
7453:
7438:
7435:
7384:
7205:
7190:
7187:
7166:{\displaystyle \mathbf {R} ^{n}}
7153:
7107:
7068:{\displaystyle \mathbf {R} ^{n}}
7055:
7003:
6988:
6985:
6960:
6734:
6691:
6668:
6599:
6332:
6329:
6200:
6165:
6130:in the space of real or complex
6107:
6092:
6089:
6058:
6043:
6040:
6012:
5977:
5827:
5824:
5791:orthogonal projection operators
5742:
5739:
5181:
5178:
3594:on the coordinate neighbourhood
3508:
3505:
2963:
2960:
1798:
1795:
1564:
1561:
1438:
1435:
1228:
1225:
1127:
929:
914:
911:
867:{\displaystyle \mathbf {R} ^{n}}
854:
838:{\displaystyle \mathbf {R} ^{n}}
825:
759:{\displaystyle \mathbf {R} ^{n}}
746:
569:
566:
529:
526:
488:
485:
448:
445:
398:
383:
380:
318:vector space, Grassmannians are
247:
187:
184:
45:
42:
21012:
20983:
20930:
20904:
20851:
20806:
20731:
20508:American Journal of Mathematics
20499:
20455:American Journal of Mathematics
20446:
20418:Journal of Mathematical Physics
20379:Journal of Mathematical Physics
20001:Kadomtsev–Petviashvili equation
19968:
19941:Maximal isotropic Grassmannians
17873:
16623:-dimensional subspace into its
12300:{\displaystyle \mathbf {P} (V)}
6575:) as affine algebraic varieties
5593:
5571:
5430:into the orthogonal direct sum
1863:{\displaystyle {\tilde {W}}=Wg}
1593:consist of the elements of the
263:{\displaystyle \mathbf {P} (V)}
21025:"A-homotopy theory of schemes"
20939:Journal of High Energy Physics
20815:Studies in Applied Mathematics
20244:
20209:
20197:
19890:
19878:
19804:
19798:
19777:
19765:
19725:
19713:
19602:
19599:
19587:
19575:
19569:
19560:
19552:
19546:
19534:
19519:
19469:
19454:
19410:
19395:
19257:
19254:
19248:
19230:
19188:
19185:
19179:
19173:
19143:
19137:
19103:
19097:
19075:
19066:
19025:
19019:
18970:
18964:
18933:
18927:
18886:
18880:
18839:
18833:
18792:
18786:
18729:
18723:
18565:
18555:
18549:
18543:
18524:
18518:
18464:
18458:
18452:
18446:
18271:
18259:
18168:
18153:
17839:
17829:
17754:
17733:
17678:
17657:
17600:
17585:
17563:and consider the partition of
17483:
17468:
17369:
17325:
17314:
17309:
17303:
17273:
17261:
17239:have the following dimensions
17189:
17183:
17142:
17136:
17115:
17103:
17067:
17054:
16929:
16921:
16892:
16860:
16822:
16809:
16709:
16703:
16677:, which were first applied in
16642:
16630:
16543:
16537:
16496:
16490:
16337:
16311:
16299:
16296:
16290:
16223:
16211:
16142:
16129:
16116:
16107:
16092:
16089:
16002:
15987:
15951:
15937:
15931:
15925:
15842:
15830:
15756:
15740:
15715:
15712:
15706:
15688:
15566:
15488:
15468:
15452:
15441:
15435:
15417:
15362:
15356:
15297:
15291:
14966:
14956:
14904:
14898:
14863:
14860:
14854:
14836:
14618:For any two ordered sequences
14602:
14586:
14553:. Since a change of basis for
14520:
14488:
14418:
14412:
14369:
14337:
14317:
14311:
14161:
14115:
13827:
13800:
13742:
13710:
13643:
13605:
13585:
13553:
13447:
13415:
13355:
13339:
13314:
13308:
13246:
13240:
13176:
13170:
13076:
13068:
13045:
13013:
13007:
13001:
12978:
12972:
12929:
12897:
12877:
12871:
12727:
12724:
12712:
12601:
12589:
12521:
12515:
12512:
12496:
12482:
12476:
12473:
12467:
12456:
12444:
12410:
12394:
12373:
12367:
12338:
12332:
12329:
12319:
12294:
12288:
12237:
12231:
12228:
12212:
12170:{\displaystyle {\mathcal {E}}}
12076:
12066:
12021:
12015:
12012:
11996:
11865:
11855:
11847:
11844:
11827:
11789:
11773:
11762:
11722:
11706:
11685:
11675:
11658:
11642:
11613:
11610:
11600:
11566:
11550:
11500:
11484:
11453:{\displaystyle {\mathcal {G}}}
11413:
11397:
11347:{\displaystyle {\mathcal {G}}}
11309:
11303:
11277:
11274:
11268:
11232:
11198:
11181:
11150:
11147:
11141:
11056:
11028:
10996:
10980:
10902:
10890:
10824:{\displaystyle {\mathcal {E}}}
10760:{\displaystyle {\mathcal {E}}}
10726:
10710:
10655:
10632:
10404:{\displaystyle {\mathcal {E}}}
10316:
10304:
10295:
10289:
10273:
10267:
10258:
10237:
10103:
10074:
10055:
10046:
10029:
10008:
9992:
9980:
9971:
9959:
9928:
9916:
9853:
9841:
9832:
9826:
9810:
9804:
9795:
9774:
9622:
9605:
9584:
9575:
9565:
9551:
9535:
9523:
9514:
9502:
9430:{\displaystyle w_{0}^{\perp }}
9388:
9366:
9340:
9331:
9314:
9293:
9273:{\displaystyle w_{0}\subset V}
9214:
9202:
9154:
9142:
9007:
9001:
8930:
8924:
8865:
8859:
8749:
8743:
8729:
8717:
8686:
8674:
8643:
8637:
8606:
8600:
8570:
8557:
8549:
8544:
8538:
8515:
8502:
8376:
8370:
8333:
8320:
8288:
8276:
8228:{\displaystyle w_{0}\subset V}
8143:
8137:
8011:
7979:
7877:
7845:
7775:
7769:
7723:{\displaystyle P^{\dagger }=P}
7598:
7584:
7529:
7515:
7506:
7486:
7463:
7442:
7356:
7324:
7215:
7194:
7111:
7091:
7013:
6992:
6964:
6944:
6911:
6905:
6738:
6724:
6695:
6681:
6672:
6652:
6603:
6589:
6512:-dimensional subspaces, where
6397:
6380:
6348:
6336:
6292:
6276:
6262:
6256:
6117:
6096:
6068:
6047:
5930:
5924:
5877:
5871:
5843:
5831:
5758:
5746:
5714:
5708:
5554:
5548:
5499:and its orthogonal complement
5364:
5197:
5191:
4970:
4804:
4760:
4690:
4680:
4642:
4594:
4377:
4293:
4010:
3978:
3918:
3878:
3834:
3796:
3784:
3524:
3518:
3477:
3471:
3405:
3373:
3230:
3218:
2979:
2973:
2770:
2738:
2501:
2485:
2439:
2427:
2168:
2156:
2117:(which amounts to multiplying
2093:homogeneous coordinate matrix
2051:
2045:
1906:
1894:
1845:
1814:
1808:
1764:
1580:
1574:
1529:
1497:
1454:
1448:
1367:
1335:
1244:
1238:
939:
924:
598:to denote the Grassmannian of
585:
573:
545:
539:
504:
492:
464:
458:
408:
393:
347:
335:
257:
251:
203:
197:
61:
55:
1:
21232:Graduate Texts in Mathematics
21145:Vector Bundles & K-Theory
21086:
20574:10.1090/S0894-0347-09-00640-7
19738:is defined as the subvariety
19302:{\displaystyle \gamma _{k,n}}
17966:{\displaystyle \chi _{k,n}=0}
16421:. Choosing an isomorphism of
15769:under the Plücker embedding.
15333:{\displaystyle \Lambda ^{k}V}
14481:space generated by the basis
14474:{\displaystyle \Lambda ^{k}V}
12821:-dimensional subspace of the
12660:{\displaystyle \Lambda ^{k}V}
12197:, the set of rational points
11981:. Therefore, the elements of
11074:In particular, for any point
7369:is any orthonormal basis for
6485:{\displaystyle w,w'\subset V}
3808:{\displaystyle (n-k)\times k}
3242:{\displaystyle (n-k)\times k}
2451:{\displaystyle (n-k)\times k}
1081:maps to the Grassmannian are
697:. Suppose we have a manifold
689:A natural example comes from
661:
618:-dimensional subspaces of an
21:Grassmannian (disambiguation)
21341:Algebraic homogeneous spaces
20738:Sato, Mikio (October 1981).
20589:Kokyuroku, RIMS, Kyoto Univ.
20204:Milnor & Stasheff (1974)
20033:in maximally supersymmetric
15279:projective algebraic variety
10767:is finitely generated. When
10363:The Grassmannian as a scheme
9082:{\displaystyle \mathbf {R} }
9060:{\displaystyle \mathbf {C} }
9038:{\displaystyle \mathbf {R} }
8837:{\displaystyle \mathbf {C} }
8815:{\displaystyle \mathbf {R} }
5224:Milnor & Stasheff (1974)
2174:{\displaystyle g\in GL(k,K)}
2115:elementary column operations
1912:{\displaystyle g\in GL(k,K)}
1773:{\displaystyle {\tilde {W}}}
1724:{\displaystyle i=1,\dots ,k}
362:projective algebraic variety
270:of one dimension lower than
7:
21205:Helgason, Sigurdur (1978),
20194:, p. 42, Example 1.24.
20182:, p. 22, Example 1.36.
20055:
19373:. It is a double cover of
18762:{\displaystyle \theta _{n}}
18476:{\displaystyle c(E)c(F)=1.}
17439:{\displaystyle \chi _{k,n}}
17155:consists of those elements
14824:Plücker-Grassmann relations
13054:{\displaystyle \iota (w)=,}
12177:is given by a vector space
12133:is the spectrum of a field
10831:is given by a vector space
10787:is the spectrum of a field
8800:If the underlying field is
2183:reduced column echelon form
1780:represent the same element
236:, so it is the same as the
10:
21362:
21291:Basic Algebraic Geometry 1
20142:homotopy theory of schemes
19856:{\displaystyle w\subset V}
19344:-dimensional subspaces of
18742:in the following way. Let
17617:-dimensional subspaces of
17500:-dimensional subspaces of
16374:{\displaystyle W\subset V}
16236:-dimensional subspaces of
16184:-dimensional subspaces of
15906:. This gives the natural
15816:{\displaystyle W\subset V}
15772:
14082:be the determinant of the
13499:{\displaystyle w\subset V}
13193:of the set of all vectors
12841:-dimensional vector space
12794:{\displaystyle w\subset V}
12563:
11217:to the usual Grassmannian
8659:We may therefore identify
8434:{\displaystyle \{h\in H\}}
8339:{\displaystyle w=g(w_{0})}
8182:-dimensional subspaces of
7424:component column vectors.
7046:-dimensional subspaces of
5519:{\displaystyle w^{\perp }}
5403:{\displaystyle w\subset V}
2137:by a sequence of elements
1426:, viewed as an element of
1419:{\displaystyle w\subset V}
638:-dimensional vector space
18:
21304:10.1007/978-3-642-37956-7
20890:10.1007/s00222-014-0506-3
20541:Mukhin, E.; Tarasov, V.;
19149:{\displaystyle g\in O(n)}
17691:and the latter is a rank
17232:{\displaystyle \{V_{i}\}}
17079:{\displaystyle (n-k)^{k}}
16681:. The Schubert cells for
16014:{\displaystyle (V/W)^{*}}
15481:under the Plücker map as
15374:{\displaystyle \dim(V)=4}
15303:{\displaystyle \iota (w)}
14424:{\displaystyle \iota (w)}
14101:{\displaystyle k\times k}
13951:For any ordered sequence
13526:-dimensional subspace of
13182:{\displaystyle \iota (w)}
12984:{\displaystyle \iota (w)}
12883:{\displaystyle \iota (w)}
11114:, the canonical morphism
10435:. Fix a positive integer
8958:that the Grassmannian is
7561:{\displaystyle n\times n}
7228:to the projection matrix
7144:-dimensional subspace of
6635:{\displaystyle n\times n}
6616:denote the space of real
6149:{\displaystyle n\times n}
4902:{\displaystyle k\times k}
3964:{\displaystyle k\times k}
3490:representing the element
3437:{\displaystyle n\times k}
3031:{\displaystyle k\times k}
2671:{\displaystyle k\times k}
2534:. In general, the first
2458:affine coordinate matrix
2086:{\displaystyle n\times k}
1991:{\displaystyle n\times k}
1952:matrices with entries in
1945:{\displaystyle k\times k}
1612:{\displaystyle n\times k}
20860:Inventiones Mathematicae
20669:10.2977/prims/1195182017
20167:
20134:classifying space for U(
20109:Lagrangian Grassmannians
11354:, which is an object of
10617:. We denote this set by
9227:and the stabiliser of a
8699:with the quotient space
6642:matrices and the subset
2507:{\displaystyle (a_{ij})}
20969:10.1007/JHEP10(2014)030
20217:Grothendieck, Alexander
20120:Lagrangian Grassmannian
20035:super Yang-Mills theory
19837:-dimensional subspaces
18492:ring was calculated by
15980:yields an inclusion of
15728:of the Grassmannian in
14029:positive integers, let
11515:, locally free of rank
9187:-dimensional subspaces
8349:for some group element
7664:{\displaystyle P^{2}=P}
6863:{\displaystyle P^{T}=P}
6804:{\displaystyle P^{2}=P}
5281:, depending on whether
2945:be the set of elements
1544:homogeneous coordinates
674:choice of subspaces or
80:differentiable manifold
21190:. New York: Springer.
21166:Characteristic classes
20338:Cartan, Élie (1981) .
20124:Grassmannians provide
20105:Isotropic Grassmanians
20011:are equivalent to the
19929:
19857:
19831:
19811:
19732:
19680:
19660:
19640:
19612:
19476:
19417:
19367:
19338:
19303:
19270:
19201:
19150:
19110:
19032:
18998:
18893:
18846:
18799:
18763:
18736:
18695:
18675:
18647:
18620:
18584:
18477:
18417:
18397:
18373:
18353:
18330:
18298:
18278:
18244:which generalizes the
18238:
18215:
18195:
18175:
18112:
18011:
17987:
17967:
17919:
17761:
17705:
17685:
17607:
17557:
17526:-dimensional subspace
17520:
17490:
17440:
17388:
17233:
17196:
17149:
17080:
17034:
16971:
16956:
16899:
16835:
16785:
16716:
16649:
16617:
16597:
16570:
16550:
16503:
16462:
16435:
16415:
16375:
16344:
16257:
16230:
16198:
16178:
16152:
16069:
16042:
16015:
15978:linear transformations
15961:
15900:
15880:
15849:
15817:
15797:-dimensional subspace
15791:
15763:
15722:
15667:
15573:
15475:
15401:
15375:
15334:
15304:
15271:
15244:
15192:
15110:
14955:
14911:
14870:
14812:
14786:
14760:
14686:
14609:
14567:
14547:
14527:
14475:
14445:
14425:
14396:
14376:
14324:
14273:
14168:
14102:
14076:
14023:
14003:
13940:
13769:
13749:
13697:
13677:
13650:
13592:
13540:
13520:
13500:
13474:
13454:
13402:
13382:
13362:
13321:
13259:
13213:
13212:{\displaystyle v\in V}
13183:
13154:
13134:
13133:{\displaystyle \iota }
13106:
13092:A different basis for
13083:
13055:
12985:
12956:
12936:
12884:
12855:
12835:
12815:
12795:
12764:
12681:
12661:
12628:
12608:
12548:
12416:
12345:
12301:
12270:
12244:
12191:
12171:
12147:
12127:
12099:
12048:
12028:
11975:
11955:
11924:
11888:
11799:
11732:
11573:
11529:
11509:
11454:
11430:
11348:
11316:
11284:
11211:
11160:
11108:
11088:
11063:
10954:
10929:
10909:
10869:
10845:
10825:
10801:
10781:
10761:
10733:
10689:
10662:
10611:
10591:
10566:
10489:
10469:
10449:
10429:
10405:
10375:by expressing it as a
10331:
10213:
10156:
10118:
9935:
9934:{\displaystyle U(V,h)}
9897:
9868:
9750:
9730:
9673:
9634:
9478:
9458:
9431:
9395:
9274:
9241:
9221:
9181:
9161:
9160:{\displaystyle O(V,q)}
9123:
9103:
9083:
9061:
9039:
9014:
8976:
8937:
8903:
8872:
8838:
8816:
8791:
8764:
8693:
8650:
8613:
8466:
8435:
8403:
8383:
8340:
8295:
8249:
8229:
8196:
8176:
8150:
8102:
8069:
8018:
7966:
7937:-dimensional subspace
7931:
7911:
7884:
7832:
7790:
7746:
7724:
7687:
7665:
7628:
7605:
7562:
7536:
7470:
7418:
7398:
7363:
7306:
7273:
7222:
7167:
7138:
7118:
7069:
7040:
7020:
6971:
6924:
6886:
6864:
6827:
6805:
6768:
6745:
6702:
6636:
6610:
6538:
6506:
6486:
6444:
6355:
6308:
6220:
6185:
6150:
6124:
6075:
6026:
5991:
5962:In particular, taking
5951:
5805:
5785:
5765:
5721:
5674:
5654:
5629:
5520:
5493:
5468:
5424:
5404:
5374:
5335:
5321:-dimensional subspace
5315:
5301:is real or complex. A
5295:
5275:
5255:
5204:
5163:
5065:
5012:
4950:
4923:
4903:
4877:
4779:
4569:
4516:
4458:
4268:
4215:
4162:
4070:
4017:
3965:
3934:
3809:
3771:
3751:
3694:
3641:
3584:
3531:
3484:
3458:
3438:
3412:
3360:
3296:
3243:
3205:
3152:
3132:
3105:
3085:
3032:
3006:
2986:
2939:
2886:
2825:
2797:
2777:
2725:
2672:
2646:
2588:
2568:
2548:
2528:
2508:
2472:
2452:
2409:
2219:
2199:
2175:
2131:
2107:
2087:
2058:
2032:
2012:
1992:
1966:
1946:
1913:
1864:
1821:
1774:
1745:
1725:
1687:
1660:
1640:
1613:
1587:
1536:
1481:
1461:
1420:
1400:-dimensional subspace
1394:
1374:
1322:
1295:
1275:
1251:
1147:
1063:
1043:
1012:
998:, and hence defines a
992:
972:
971:{\displaystyle x\in M}
946:
892:
868:
839:
810:
786:
785:{\displaystyle x\in M}
760:
731:
711:
652:
632:
612:
592:
552:
511:
471:
415:
354:
353:{\displaystyle k(n-k)}
304:
284:
264:
230:
210:
169:
146:
123:
96:
68:
21331:Differential geometry
21106:John Wiley & Sons
20782:10.1017/9781108610902
20772:; Balogh, F. (2021).
20707:10.1017/9781108610902
20697:; Balogh, F. (2021).
20341:The theory of spinors
20276:John Wiley & Sons
20115:Isotropic Grassmanian
20068:differential geometry
20027:scattering amplitudes
19959:Plücker map embedding
19930:
19858:
19832:
19812:
19733:
19681:
19661:
19641:
19613:
19477:
19418:
19368:
19339:
19317:Oriented Grassmannian
19304:
19271:
19202:
19151:
19111:
19033:
18999:
18894:
18847:
18800:
18764:
18737:
18696:
18676:
18648:
18621:
18585:
18478:
18418:
18398:
18374:
18354:
18331:
18299:
18279:
18277:{\displaystyle (n-k)}
18239:
18216:
18196:
18176:
18113:
18012:
17988:
17968:
17920:
17762:
17706:
17686:
17608:
17558:
17521:
17491:
17441:
17389:
17234:
17197:
17150:
17081:
17035:
16972:
16936:
16900:
16836:
16786:
16717:
16657:orthogonal complement
16650:
16648:{\displaystyle (n-k)}
16618:
16598:
16596:{\displaystyle V^{*}}
16571:
16556:. An isomorphism of
16551:
16504:
16463:
16461:{\displaystyle V^{*}}
16436:
16416:
16376:
16345:
16258:
16256:{\displaystyle V^{*}}
16231:
16229:{\displaystyle (n-k)}
16199:
16179:
16153:
16070:
16068:{\displaystyle W^{*}}
16043:
16041:{\displaystyle V^{*}}
16016:
15962:
15901:
15881:
15850:
15848:{\displaystyle (n-k)}
15818:
15792:
15764:
15723:
15668:
15574:
15476:
15402:
15376:
15335:
15305:
15272:
15270:{\displaystyle j_{l}}
15245:
15198:denotes the sequence
15193:
15111:
14929:
14912:
14871:
14813:
14787:
14761:
14687:
14610:
14568:
14548:
14528:
14476:
14446:
14426:
14397:
14377:
14325:
14274:
14169:
14103:
14077:
14024:
14004:
13941:
13770:
13750:
13698:
13678:
13676:{\displaystyle w_{i}}
13656:be the components of
13651:
13593:
13541:
13521:
13501:
13475:
13455:
13403:
13388:th exterior power of
13383:
13363:
13322:
13260:
13214:
13184:
13155:
13135:
13107:
13084:
13056:
12986:
12957:
12937:
12885:
12856:
12836:
12816:
12796:
12765:
12682:
12662:
12629:
12609:
12560:The Plücker embedding
12549:
12417:
12346:
12302:
12271:
12245:
12192:
12172:
12148:
12128:
12100:
12049:
12029:
11976:
11956:
11925:
11923:{\displaystyle O_{T}}
11889:
11800:
11733:
11574:
11530:
11510:
11455:
11431:
11349:
11317:
11285:
11212:
11161:
11109:
11089:
11064:
10955:
10930:
10910:
10870:
10846:
10826:
10802:
10782:
10762:
10734:
10690:
10663:
10612:
10592:
10577:locally free of rank
10567:
10490:
10470:
10450:
10430:
10406:
10383:Representable functor
10377:representable functor
10332:
10214:
10157:
10119:
9936:
9898:
9869:
9751:
9731:
9674:
9635:
9479:
9459:
9457:{\displaystyle w_{0}}
9432:
9396:
9275:
9242:
9222:
9182:
9162:
9124:
9104:
9084:
9062:
9040:
9015:
8977:
8938:
8904:
8873:
8839:
8817:
8792:
8765:
8694:
8651:
8614:
8467:
8465:{\displaystyle w_{0}}
8436:
8404:
8384:
8341:
8296:
8250:
8230:
8197:
8177:
8151:
8103:
8049:
8019:
7967:
7932:
7912:
7910:{\displaystyle P_{w}}
7885:
7833:
7791:
7747:
7725:
7688:
7666:
7629:
7606:
7563:
7537:
7471:
7419:
7399:
7364:
7307:
7253:
7223:
7168:
7139:
7119:
7070:
7041:
7021:
6977:and the Grassmannian
6972:
6925:
6887:
6865:
6828:
6806:
6769:
6746:
6703:
6637:
6611:
6539:
6507:
6487:
6445:
6356:
6309:
6221:
6186:
6151:
6125:
6076:
6027:
5992:
5952:
5806:
5786:
5771:with the set of rank
5766:
5722:
5675:
5655:
5630:
5521:
5494:
5469:
5425:
5405:
5375:
5336:
5316:
5296:
5276:
5256:
5205:
5164:
5066:
5013:
4951:
4949:{\displaystyle j_{l}}
4924:
4904:
4878:
4780:
4570:
4517:
4459:
4269:
4216:
4163:
4071:
4018:
3966:
3935:
3810:
3772:
3752:
3695:
3642:
3585:
3532:
3485:
3459:
3439:
3413:
3361:
3297:
3244:
3206:
3153:
3133:
3131:{\displaystyle i_{j}}
3106:
3086:
3033:
3007:
2987:
2940:
2887:
2826:
2798:
2778:
2726:
2673:
2647:
2589:
2569:
2549:
2529:
2509:
2473:
2453:
2410:
2220:
2200:
2176:
2132:
2108:
2088:
2059:
2033:
2013:
1993:
1967:
1947:
1914:
1865:
1822:
1775:
1746:
1726:
1688:
1686:{\displaystyle W_{i}}
1666:-th column vector is
1661:
1641:
1614:
1588:
1537:
1482:
1462:
1421:
1395:
1375:
1323:
1321:{\displaystyle K^{n}}
1296:
1276:
1252:
1148:
1064:
1044:
1013:
993:
973:
947:
893:
869:
845:, which is also just
840:
811:
787:
761:
732:
712:
684:differential manifold
653:
633:
613:
593:
553:
512:
472:
416:
355:
305:
285:
265:
231:
211:
170:
147:
124:
97:
69:
21286:Shafarevich, Igor R.
21184:Harris, Joe (1992).
20626:10.1143/jpsj.50.3806
20086:Plücker co-ordinates
19872:
19841:
19821:
19742:
19690:
19670:
19650:
19630:
19492:
19427:
19377:
19348:
19328:
19280:
19211:
19160:
19125:
19047:
19031:{\displaystyle O(n)}
19013:
18908:
18856:
18809:
18798:{\displaystyle O(n)}
18780:
18746:
18705:
18685:
18665:
18634:
18607:
18597:quantum field theory
18505:
18440:
18407:
18387:
18363:
18343:
18320:
18288:
18256:
18228:
18205:
18185:
18135:
18030:
18001:
17977:
17938:
17776:
17715:
17695:
17633:
17567:
17530:
17510:
17450:
17446:of the Grassmannian
17417:
17412:Euler characteristic
17248:
17210:
17159:
17090:
17086:, the Schubert cell
17051:
16989:
16917:
16851:
16795:
16730:
16685:
16679:enumerative geometry
16627:
16607:
16580:
16560:
16513:
16472:
16445:
16425:
16385:
16359:
16272:
16240:
16208:
16188:
16168:
16083:
16052:
16025:
15984:
15919:
15908:short exact sequence
15890:
15862:
15827:
15801:
15781:
15732:
15682:
15588:
15485:
15411:
15385:
15347:
15314:
15285:
15254:
15202:
15125:
14926:
14880:
14830:
14796:
14770:
14696:
14622:
14578:
14557:
14537:
14485:
14455:
14435:
14406:
14386:
14334:
14287:
14178:
14112:
14108:matrix with columns
14086:
14033:
14013:
13955:
13784:
13759:
13707:
13687:
13660:
13602:
13550:
13530:
13510:
13484:
13464:
13412:
13392:
13372:
13331:
13290:
13228:
13197:
13164:
13144:
13124:
13096:
13065:
12995:
12966:
12946:
12894:
12865:
12845:
12825:
12805:
12779:
12696:
12671:
12641:
12618:
12578:
12436:
12360:
12311:
12280:
12254:
12201:
12181:
12157:
12137:
12111:
12058:
12038:
11985:
11965:
11934:
11907:
11819:
11756:
11743:For any morphism of
11592:
11539:
11519:
11464:
11440:
11358:
11334:
11315:{\displaystyle K(s)}
11297:
11221:
11170:
11118:
11098:
11078:
10969:
10939:
10919:
10879:
10859:
10835:
10811:
10791:
10771:
10747:
10699:
10679:
10621:
10601:
10581:
10508:
10479:
10459:
10439:
10419:
10413:quasi-coherent sheaf
10391:
10226:
10166:
10131:
9948:
9910:
9887:
9763:
9740:
9683:
9648:
9491:
9468:
9441:
9409:
9287:
9251:
9231:
9191:
9171:
9136:
9113:
9093:
9071:
9049:
9027:
8990:
8966:
8913:
8893:
8848:
8826:
8804:
8781:
8706:
8663:
8626:
8479:
8449:
8413:
8393:
8353:
8308:
8301:can be expressed as
8259:
8239:
8206:
8186:
8166:
8126:
8033:
7976:
7941:
7921:
7894:
7842:
7808:
7756:
7736:
7701:
7677:
7642:
7618:
7572:
7546:
7480:
7431:
7408:
7373:
7321:
7237:
7177:
7148:
7128:
7079:
7050:
7030:
6981:
6938:
6896:
6876:
6841:
6817:
6782:
6758:
6712:
6646:
6620:
6583:
6528:
6496:
6459:
6374:
6325:
6233:
6195:
6160:
6134:
6085:
6036:
6001:
5966:
5820:
5795:
5775:
5735:
5684:
5664:
5644:
5535:
5503:
5483:
5439:
5414:
5388:
5345:
5325:
5305:
5285:
5265:
5233:
5220:projection operators
5173:
5075:
5022:
4960:
4933:
4913:
4887:
4794:
4584:
4526:
4473:
4283:
4225:
4172:
4080:
4027:
3975:
3971:submatrix with rows
3949:
3824:
3815:matrices. Denote by
3781:
3761:
3708:
3651:
3598:
3541:
3494:
3468:
3448:
3422:
3370:
3306:
3253:
3215:
3162:
3142:
3115:
3095:
3042:
3016:
2996:
2949:
2896:
2838:
2815:
2787:
2735:
2731:whose rows are the
2682:
2656:
2598:
2578:
2558:
2538:
2518:
2482:
2462:
2424:
2234:
2209:
2189:
2141:
2121:
2097:
2071:
2042:
2022:
2002:
1976:
1956:
1930:
1921:general linear group
1879:
1836:
1784:
1755:
1735:
1697:
1670:
1650:
1630:
1597:
1550:
1494:
1488:linearly independent
1471:
1430:
1404:
1384:
1332:
1305:
1285:
1265:
1220:
1122:
1053:
1033:
1002:
982:
956:
906:
882:
849:
820:
800:
770:
741:
721:
701:
642:
622:
602:
562:
521:
481:
440:
375:
329:
294:
274:
243:
220:
179:
159:
136:
113:
86:
74:(named in honour of
37:
19:For other uses, see
21336:Projective geometry
21021:Voevodsky, Vladimir
20961:2014JHEP...10..030A
20882:2014InMat.198..637K
20618:1981JPSJ...50.3806D
20551:J. Amer. Math. Soc
20430:1995JMP....36.1945H
20391:1992JMP....33.3197H
20152:Affine Grassmannian
20078:projective geometry
20031:subatomic particles
20003:and the associated
19764:
19712:
19666:-dimensional space
18429:total Chern classes
18246:tautological bundle
17934:gives the formula:
17932:recursion relations
17711:vector bundle over
17206:with the subspaces
15879:{\displaystyle V/W}
15400:{\displaystyle k=2}
14811:{\displaystyle k+1}
14785:{\displaystyle k-1}
14281:Plücker coordinates
12307:, and the image of
12269:{\displaystyle k-1}
12126:{\displaystyle T=S}
10100:
10072:
9426:
9385:
9357:
8878:is considered as a
8094:
7636:projection operator
7298:
6776:projection operator
5680:defines a subspace
5169:gives an atlas for
4872:
4758:
3355:
1100:, the Grassmannian
423:Plücker coordinates
21346:Algebraic geometry
21209:, Academic Press,
21162:Stasheff, James D.
21094:Griffiths, Phillip
21046:10.1007/BF02698831
20476:10338.dmlcz/700905
20346:Dover Publications
20264:Griffiths, Phillip
20146:algebraic K-theory
20126:classifying spaces
19984:. Subvarieties of
19982:Schubert varieties
19925:
19853:
19827:
19817:consisting of all
19807:
19745:
19728:
19693:
19676:
19656:
19636:
19608:
19472:
19423:and is denoted by
19413:
19363:
19334:
19299:
19266:
19197:
19146:
19106:
19028:
18994:
18889:
18842:
18795:
18759:
18732:
18691:
18671:
18657:Associated measure
18646:{\displaystyle 2n}
18643:
18619:{\displaystyle 2n}
18616:
18580:
18490:quantum cohomology
18473:
18413:
18393:
18369:
18349:
18326:
18294:
18274:
18234:
18211:
18191:
18171:
18108:
18102:
18007:
17983:
17963:
17915:
17757:
17701:
17681:
17603:
17553:
17516:
17486:
17436:
17405:Schubert varieties
17384:
17229:
17192:
17145:
17076:
17030:
16967:
16895:
16831:
16781:
16712:
16645:
16613:
16593:
16566:
16546:
16499:
16458:
16431:
16411:
16371:
16340:
16253:
16226:
16194:
16174:
16148:
16065:
16038:
16011:
15957:
15896:
15876:
15845:
15813:
15787:
15759:
15718:
15663:
15569:
15471:
15397:
15371:
15330:
15300:
15267:
15240:
15188:
15106:
14907:
14866:
14808:
14782:
14756:
14682:
14605:
14563:
14543:
14523:
14471:
14441:
14421:
14392:
14372:
14320:
14269:
14164:
14098:
14072:
14019:
13999:
13936:
13927:
13765:
13745:
13693:
13673:
13646:
13588:
13536:
13516:
13496:
13470:
13450:
13398:
13378:
13358:
13317:
13255:
13209:
13179:
13150:
13130:
13102:
13079:
13051:
12981:
12952:
12932:
12880:
12851:
12831:
12811:
12791:
12677:
12657:
12624:
12604:
12544:
12341:
12297:
12266:
12240:
12187:
12167:
12143:
12123:
12095:
12044:
12024:
11971:
11951:
11920:
11884:
11795:
11728:
11569:
11525:
11505:
11450:
11426:
11344:
11312:
11280:
11207:
11156:
11104:
11084:
11059:
10953:{\displaystyle S'}
10950:
10925:
10905:
10865:
10841:
10821:
10797:
10777:
10757:
10729:
10685:
10658:
10607:
10587:
10562:
10485:
10465:
10445:
10425:
10401:
10369:algebraic geometry
10327:
10209:
10152:
10114:
10086:
10058:
9931:
9893:
9864:
9746:
9726:
9669:
9630:
9474:
9454:
9427:
9412:
9391:
9371:
9343:
9270:
9237:
9217:
9177:
9157:
9119:
9099:
9079:
9057:
9035:
9010:
8984:parabolic subgroup
8972:
8933:
8899:
8868:
8834:
8812:
8787:
8760:
8689:
8646:
8609:
8462:
8431:
8399:
8379:
8336:
8291:
8245:
8225:
8192:
8172:
8146:
8098:
8080:
8014:
7962:
7927:
7907:
7880:
7828:
7786:
7742:
7720:
7683:
7661:
7624:
7601:
7558:
7532:
7466:
7414:
7394:
7359:
7302:
7284:
7218:
7163:
7134:
7114:
7065:
7036:
7016:
6967:
6920:
6882:
6860:
6823:
6801:
6764:
6741:
6698:
6632:
6606:
6534:
6502:
6482:
6440:
6351:
6304:
6216:
6181:
6146:
6120:
6071:
6022:
5987:
5947:
5801:
5781:
5761:
5717:
5670:
5650:
5625:
5620:
5516:
5489:
5464:
5420:
5400:
5370:
5331:
5311:
5291:
5271:
5251:
5200:
5159:
5061:
5008:
4946:
4919:
4899:
4883:is the invertible
4873:
4797:
4775:
4683:
4565:
4512:
4454:
4264:
4211:
4158:
4066:
4013:
3961:
3930:
3805:
3767:
3747:
3690:
3637:
3580:
3527:
3480:
3454:
3444:coordinate matrix
3434:
3408:
3356:
3312:
3292:
3239:
3201:
3148:
3128:
3101:
3081:
3028:
3002:
2982:
2935:
2882:
2821:
2793:
2773:
2721:
2668:
2642:
2584:
2574:has maximal rank
2564:
2544:
2524:
2504:
2468:
2448:
2405:
2399:
2215:
2195:
2171:
2127:
2103:
2083:
2054:
2028:
2008:
1988:
1962:
1942:
1909:
1860:
1817:
1770:
1741:
1721:
1683:
1656:
1636:
1609:
1583:
1532:
1477:
1457:
1416:
1390:
1370:
1318:
1291:
1271:
1247:
1143:
1059:
1039:
1008:
988:
968:
942:
888:
864:
835:
806:
782:
756:
727:
707:
648:
628:
608:
588:
548:
507:
467:
411:
350:
300:
280:
260:
226:
206:
165:
142:
119:
92:
64:
16:Mathematical space
21313:978-0-387-97716-4
21245:978-1-4419-9981-8
21216:978-0-8218-2848-9
20999:978-1-4244-2242-5
20355:978-0-486-64070-9
20236:978-3-540-05113-8
20098:Stiefel manifolds
20082:Plücker embedding
20062:Schubert calculus
20013:Plücker relations
19978:Schubert calculus
19830:{\displaystyle k}
19679:{\displaystyle V}
19659:{\displaystyle n}
19639:{\displaystyle Q}
19510:
19445:
19337:{\displaystyle k}
18852:. Then for a set
18694:{\displaystyle n}
18674:{\displaystyle V}
18416:{\displaystyle F}
18396:{\displaystyle E}
18372:{\displaystyle F}
18352:{\displaystyle E}
18329:{\displaystyle E}
18297:{\displaystyle F}
18252:. Similarly the
18237:{\displaystyle E}
18214:{\displaystyle n}
18194:{\displaystyle k}
18094:
18072:
18010:{\displaystyle k}
17986:{\displaystyle n}
17704:{\displaystyle k}
17519:{\displaystyle 1}
16843:integer partition
16616:{\displaystyle k}
16569:{\displaystyle V}
16434:{\displaystyle V}
16197:{\displaystyle V}
16177:{\displaystyle k}
15899:{\displaystyle V}
15790:{\displaystyle k}
15163:
15070:
14820:Plücker relations
14566:{\displaystyle w}
14546:{\displaystyle V}
14444:{\displaystyle w}
14395:{\displaystyle V}
14022:{\displaystyle k}
13768:{\displaystyle k}
13696:{\displaystyle V}
13539:{\displaystyle V}
13519:{\displaystyle k}
13473:{\displaystyle V}
13401:{\displaystyle V}
13381:{\displaystyle k}
13283:Plücker relations
13278:Plücker embedding
13153:{\displaystyle w}
13105:{\displaystyle w}
12955:{\displaystyle w}
12890:, choose a basis
12854:{\displaystyle V}
12834:{\displaystyle n}
12814:{\displaystyle k}
12680:{\displaystyle V}
12627:{\displaystyle k}
12572:Plücker embedding
12566:Plücker embedding
12190:{\displaystyle V}
12146:{\displaystyle K}
12047:{\displaystyle k}
11974:{\displaystyle k}
11528:{\displaystyle k}
11136:
11107:{\displaystyle S}
11087:{\displaystyle s}
10928:{\displaystyle S}
10868:{\displaystyle V}
10844:{\displaystyle V}
10807:, then the sheaf
10800:{\displaystyle K}
10780:{\displaystyle S}
10688:{\displaystyle S}
10610:{\displaystyle T}
10590:{\displaystyle k}
10488:{\displaystyle T}
10468:{\displaystyle S}
10448:{\displaystyle k}
10428:{\displaystyle S}
9896:{\displaystyle h}
9749:{\displaystyle k}
9477:{\displaystyle V}
9240:{\displaystyle k}
9180:{\displaystyle k}
9122:{\displaystyle V}
9102:{\displaystyle q}
8975:{\displaystyle H}
8956:Plücker embedding
8952:algebraic variety
8902:{\displaystyle K}
8790:{\displaystyle H}
8402:{\displaystyle g}
8248:{\displaystyle k}
8195:{\displaystyle V}
8175:{\displaystyle k}
8120:homogeneous space
7930:{\displaystyle k}
7917:onto the complex
7745:{\displaystyle P}
7686:{\displaystyle P}
7627:{\displaystyle P}
7417:{\displaystyle n}
7404:, viewed as real
7137:{\displaystyle k}
7075:given by sending
7039:{\displaystyle k}
6885:{\displaystyle P}
6826:{\displaystyle P}
6767:{\displaystyle P}
6537:{\displaystyle V}
6505:{\displaystyle k}
5804:{\displaystyle P}
5784:{\displaystyle k}
5673:{\displaystyle k}
5653:{\displaystyle P}
5597:
5575:
5492:{\displaystyle w}
5423:{\displaystyle V}
5334:{\displaystyle w}
5314:{\displaystyle k}
5294:{\displaystyle V}
5274:{\displaystyle V}
4973:
4922:{\displaystyle l}
4807:
4693:
4645:
4597:
4380:
4296:
3837:
3770:{\displaystyle K}
3457:{\displaystyle W}
3366:complementary to
3151:{\displaystyle W}
3104:{\displaystyle j}
3005:{\displaystyle W}
2824:{\displaystyle w}
2796:{\displaystyle W}
2587:{\displaystyle k}
2567:{\displaystyle W}
2547:{\displaystyle k}
2527:{\displaystyle w}
2471:{\displaystyle A}
2218:{\displaystyle W}
2198:{\displaystyle k}
2130:{\displaystyle W}
2106:{\displaystyle W}
2031:{\displaystyle k}
2011:{\displaystyle W}
1965:{\displaystyle K}
1875:for some element
1848:
1767:
1744:{\displaystyle W}
1659:{\displaystyle i}
1639:{\displaystyle W}
1480:{\displaystyle k}
1393:{\displaystyle k}
1294:{\displaystyle V}
1274:{\displaystyle V}
1062:{\displaystyle M}
1042:{\displaystyle M}
1011:{\displaystyle k}
991:{\displaystyle x}
891:{\displaystyle x}
809:{\displaystyle M}
730:{\displaystyle k}
710:{\displaystyle M}
651:{\displaystyle V}
631:{\displaystyle n}
611:{\displaystyle k}
431:Hermann Grassmann
303:{\displaystyle V}
283:{\displaystyle V}
229:{\displaystyle V}
168:{\displaystyle K}
145:{\displaystyle V}
122:{\displaystyle n}
95:{\displaystyle k}
76:Hermann Grassmann
21353:
21317:
21296:Springer Science
21281:
21257:
21219:
21201:
21180:see chapters 5–7
21179:
21152:
21150:
21135:
21081:
21079:
21077:
21076:
21029:
21016:
21010:
21001:, pp. 1–8 (
20987:
20981:
20980:
20954:
20934:
20928:
20927:
20925:
20923:
20908:
20902:
20901:
20875:
20855:
20849:
20848:
20830:
20810:
20804:
20803:
20766:
20760:
20759:
20735:
20729:
20728:
20691:
20682:
20681:
20671:
20647:
20638:
20637:
20601:
20592:
20585:
20579:
20578:
20576:
20566:
20538:
20532:
20531:
20503:
20497:
20496:
20478:
20450:
20444:
20443:
20441:
20439:10.1063/1.531096
20409:
20403:
20402:
20399:10.1063/1.529538
20374:
20368:
20367:
20335:
20326:
20325:
20323:
20311:
20305:
20304:
20260:
20254:
20248:
20242:
20240:
20213:
20207:
20201:
20195:
20192:Shafarevich 2013
20189:
20183:
20177:
20157:Grassmann bundle
19934:
19932:
19931:
19926:
19862:
19860:
19859:
19854:
19836:
19834:
19833:
19828:
19816:
19814:
19813:
19808:
19797:
19796:
19791:
19763:
19758:
19753:
19737:
19735:
19734:
19729:
19711:
19706:
19701:
19685:
19683:
19682:
19677:
19665:
19663:
19662:
19657:
19645:
19643:
19642:
19637:
19617:
19615:
19614:
19609:
19559:
19533:
19532:
19527:
19518:
19517:
19512:
19511:
19506:
19498:
19481:
19479:
19478:
19473:
19468:
19467:
19462:
19453:
19452:
19447:
19446:
19441:
19433:
19422:
19420:
19419:
19414:
19409:
19408:
19403:
19394:
19393:
19388:
19372:
19370:
19369:
19364:
19362:
19361:
19356:
19343:
19341:
19340:
19335:
19308:
19306:
19305:
19300:
19298:
19297:
19275:
19273:
19272:
19267:
19247:
19246:
19241:
19229:
19228:
19206:
19204:
19203:
19198:
19172:
19171:
19155:
19153:
19152:
19147:
19115:
19113:
19112:
19107:
19096:
19095:
19065:
19064:
19037:
19035:
19034:
19029:
19003:
19001:
19000:
18995:
18948:
18947:
18926:
18925:
18898:
18896:
18895:
18890:
18879:
18878:
18873:
18851:
18849:
18848:
18843:
18832:
18831:
18826:
18804:
18802:
18801:
18796:
18775:orthogonal group
18768:
18766:
18765:
18760:
18758:
18757:
18741:
18739:
18738:
18733:
18722:
18721:
18716:
18700:
18698:
18697:
18692:
18680:
18678:
18677:
18672:
18652:
18650:
18649:
18644:
18625:
18623:
18622:
18617:
18589:
18587:
18586:
18581:
18579:
18578:
18542:
18541:
18517:
18516:
18482:
18480:
18479:
18474:
18422:
18420:
18419:
18414:
18402:
18400:
18399:
18394:
18378:
18376:
18375:
18370:
18358:
18356:
18355:
18350:
18335:
18333:
18332:
18327:
18304:. The integral
18303:
18301:
18300:
18295:
18283:
18281:
18280:
18275:
18250:projective space
18243:
18241:
18240:
18235:
18220:
18218:
18217:
18212:
18200:
18198:
18197:
18192:
18180:
18178:
18177:
18172:
18167:
18166:
18161:
18152:
18151:
18146:
18117:
18115:
18114:
18109:
18107:
18106:
18099:
18095:
18087:
18077:
18073:
18065:
18048:
18047:
18016:
18014:
18013:
18008:
17992:
17990:
17989:
17984:
17972:
17970:
17969:
17964:
17956:
17955:
17924:
17922:
17921:
17916:
17908:
17907:
17889:
17888:
17869:
17868:
17847:
17846:
17825:
17824:
17794:
17793:
17766:
17764:
17763:
17758:
17753:
17752:
17741:
17732:
17731:
17726:
17710:
17708:
17707:
17702:
17690:
17688:
17687:
17682:
17677:
17676:
17665:
17656:
17655:
17644:
17628:
17622:
17616:
17612:
17610:
17609:
17604:
17599:
17598:
17593:
17584:
17583:
17578:
17562:
17560:
17559:
17554:
17552:
17551:
17546:
17537:
17525:
17523:
17522:
17517:
17505:
17499:
17495:
17493:
17492:
17487:
17482:
17481:
17476:
17467:
17466:
17461:
17445:
17443:
17442:
17437:
17435:
17434:
17401:Zariski topology
17393:
17391:
17390:
17385:
17368:
17367:
17366:
17365:
17317:
17302:
17301:
17296:
17260:
17259:
17238:
17236:
17235:
17230:
17225:
17224:
17201:
17199:
17198:
17193:
17182:
17181:
17176:
17154:
17152:
17151:
17146:
17135:
17134:
17129:
17102:
17101:
17085:
17083:
17082:
17077:
17075:
17074:
17039:
17037:
17036:
17031:
17020:
17019:
17001:
17000:
16976:
16974:
16973:
16968:
16966:
16965:
16955:
16950:
16932:
16924:
16904:
16902:
16901:
16896:
16891:
16890:
16872:
16871:
16840:
16838:
16837:
16832:
16821:
16820:
16808:
16790:
16788:
16787:
16782:
16774:
16773:
16755:
16754:
16742:
16741:
16721:
16719:
16718:
16713:
16702:
16701:
16696:
16654:
16652:
16651:
16646:
16622:
16620:
16619:
16614:
16602:
16600:
16599:
16594:
16592:
16591:
16575:
16573:
16572:
16567:
16555:
16553:
16552:
16547:
16536:
16535:
16524:
16508:
16506:
16505:
16500:
16489:
16488:
16483:
16467:
16465:
16464:
16459:
16457:
16456:
16440:
16438:
16437:
16432:
16420:
16418:
16417:
16412:
16410:
16409:
16397:
16396:
16381:its annihilator
16380:
16378:
16377:
16372:
16349:
16347:
16346:
16341:
16336:
16335:
16323:
16309:
16289:
16288:
16283:
16262:
16260:
16259:
16254:
16252:
16251:
16235:
16233:
16232:
16227:
16203:
16201:
16200:
16195:
16183:
16181:
16180:
16175:
16157:
16155:
16154:
16149:
16141:
16140:
16128:
16127:
16115:
16114:
16102:
16074:
16072:
16071:
16066:
16064:
16063:
16047:
16045:
16044:
16039:
16037:
16036:
16020:
16018:
16017:
16012:
16010:
16009:
15997:
15966:
15964:
15963:
15958:
15947:
15905:
15903:
15902:
15897:
15885:
15883:
15882:
15877:
15872:
15854:
15852:
15851:
15846:
15822:
15820:
15819:
15814:
15796:
15794:
15793:
15788:
15768:
15766:
15765:
15760:
15752:
15751:
15739:
15727:
15725:
15724:
15719:
15705:
15704:
15699:
15672:
15670:
15669:
15664:
15656:
15655:
15646:
15645:
15633:
15632:
15623:
15622:
15610:
15609:
15600:
15599:
15578:
15576:
15575:
15570:
15565:
15564:
15552:
15551:
15539:
15538:
15526:
15525:
15513:
15512:
15500:
15499:
15480:
15478:
15477:
15472:
15464:
15463:
15451:
15434:
15433:
15428:
15406:
15404:
15403:
15398:
15380:
15378:
15377:
15372:
15339:
15337:
15336:
15331:
15326:
15325:
15309:
15307:
15306:
15301:
15276:
15274:
15273:
15268:
15266:
15265:
15249:
15247:
15246:
15241:
15239:
15238:
15214:
15213:
15197:
15195:
15194:
15189:
15187:
15186:
15165:
15164:
15159:
15158:
15149:
15137:
15136:
15115:
15113:
15112:
15107:
15096:
15095:
15094:
15093:
15072:
15071:
15066:
15065:
15056:
15044:
15043:
15029:
15028:
15027:
15026:
15014:
15013:
14989:
14988:
14974:
14973:
14954:
14943:
14916:
14914:
14913:
14908:
14897:
14896:
14891:
14875:
14873:
14872:
14867:
14853:
14852:
14847:
14817:
14815:
14814:
14809:
14791:
14789:
14788:
14783:
14765:
14763:
14762:
14757:
14749:
14748:
14727:
14726:
14714:
14713:
14691:
14689:
14688:
14683:
14675:
14674:
14653:
14652:
14640:
14639:
14614:
14612:
14611:
14606:
14598:
14597:
14585:
14572:
14570:
14569:
14564:
14552:
14550:
14549:
14544:
14532:
14530:
14529:
14524:
14519:
14518:
14500:
14499:
14480:
14478:
14477:
14472:
14467:
14466:
14450:
14448:
14447:
14442:
14430:
14428:
14427:
14422:
14401:
14399:
14398:
14393:
14381:
14379:
14378:
14373:
14368:
14367:
14349:
14348:
14329:
14327:
14326:
14321:
14310:
14309:
14304:
14278:
14276:
14275:
14270:
14259:
14258:
14240:
14239:
14219:
14218:
14217:
14216:
14198:
14197:
14173:
14171:
14170:
14167:{\displaystyle }
14165:
14160:
14159:
14158:
14157:
14134:
14133:
14132:
14131:
14107:
14105:
14104:
14099:
14081:
14079:
14078:
14073:
14071:
14070:
14069:
14068:
14050:
14049:
14028:
14026:
14025:
14020:
14008:
14006:
14005:
14000:
13992:
13991:
13973:
13972:
13945:
13943:
13942:
13937:
13932:
13931:
13924:
13923:
13904:
13903:
13870:
13869:
13850:
13849:
13826:
13825:
13812:
13811:
13796:
13795:
13774:
13772:
13771:
13766:
13754:
13752:
13751:
13746:
13741:
13740:
13722:
13721:
13702:
13700:
13699:
13694:
13682:
13680:
13679:
13674:
13672:
13671:
13655:
13653:
13652:
13647:
13642:
13641:
13620:
13619:
13597:
13595:
13594:
13589:
13584:
13583:
13565:
13564:
13545:
13543:
13542:
13537:
13525:
13523:
13522:
13517:
13505:
13503:
13502:
13497:
13479:
13477:
13476:
13471:
13459:
13457:
13456:
13451:
13446:
13445:
13427:
13426:
13407:
13405:
13404:
13399:
13387:
13385:
13384:
13379:
13367:
13365:
13364:
13359:
13351:
13350:
13338:
13326:
13324:
13323:
13318:
13307:
13306:
13301:
13264:
13262:
13261:
13256:
13218:
13216:
13215:
13210:
13188:
13186:
13185:
13180:
13159:
13157:
13156:
13151:
13139:
13137:
13136:
13131:
13111:
13109:
13108:
13103:
13088:
13086:
13085:
13082:{\displaystyle }
13080:
13060:
13058:
13057:
13052:
13044:
13043:
13025:
13024:
12990:
12988:
12987:
12982:
12961:
12959:
12958:
12953:
12941:
12939:
12938:
12933:
12928:
12927:
12909:
12908:
12889:
12887:
12886:
12881:
12860:
12858:
12857:
12852:
12840:
12838:
12837:
12832:
12820:
12818:
12817:
12812:
12800:
12798:
12797:
12792:
12769:
12767:
12766:
12761:
12757:
12753:
12749:
12748:
12734:
12711:
12686:
12684:
12683:
12678:
12666:
12664:
12663:
12658:
12653:
12652:
12633:
12631:
12630:
12625:
12613:
12611:
12610:
12605:
12588:
12553:
12551:
12550:
12545:
12540:
12536:
12511:
12510:
12495:
12466:
12421:
12419:
12418:
12413:
12409:
12408:
12393:
12385:
12384:
12366:
12350:
12348:
12347:
12342:
12328:
12327:
12318:
12306:
12304:
12303:
12298:
12287:
12275:
12273:
12272:
12267:
12249:
12247:
12246:
12241:
12227:
12226:
12211:
12196:
12194:
12193:
12188:
12176:
12174:
12173:
12168:
12166:
12165:
12152:
12150:
12149:
12144:
12132:
12130:
12129:
12124:
12104:
12102:
12101:
12096:
12088:
12087:
12075:
12074:
12065:
12053:
12051:
12050:
12045:
12033:
12031:
12030:
12025:
12011:
12010:
11995:
11980:
11978:
11977:
11972:
11960:
11958:
11957:
11952:
11950:
11949:
11944:
11943:
11929:
11927:
11926:
11921:
11919:
11918:
11903:homomorphism of
11893:
11891:
11890:
11885:
11877:
11876:
11864:
11863:
11854:
11843:
11842:
11837:
11836:
11826:
11804:
11802:
11801:
11796:
11788:
11787:
11772:
11746:
11737:
11735:
11734:
11729:
11721:
11720:
11705:
11697:
11696:
11684:
11683:
11674:
11666:
11662:
11661:
11657:
11656:
11641:
11632:
11631:
11620:
11609:
11608:
11599:
11578:
11576:
11575:
11570:
11565:
11564:
11549:
11534:
11532:
11531:
11526:
11514:
11512:
11511:
11506:
11504:
11503:
11499:
11498:
11483:
11474:
11473:
11459:
11457:
11456:
11451:
11449:
11448:
11435:
11433:
11432:
11427:
11422:
11418:
11417:
11416:
11412:
11411:
11396:
11387:
11386:
11368:
11353:
11351:
11350:
11345:
11343:
11342:
11326:Universal family
11321:
11319:
11318:
11313:
11289:
11287:
11286:
11281:
11264:
11263:
11262:
11261:
11247:
11246:
11231:
11216:
11214:
11213:
11208:
11206:
11205:
11196:
11195:
11180:
11165:
11163:
11162:
11157:
11137:
11134:
11113:
11111:
11110:
11105:
11093:
11091:
11090:
11085:
11068:
11066:
11065:
11060:
11055:
11054:
11053:
11044:
11043:
11027:
11016:
11008:
11007:
10995:
10994:
10979:
10959:
10957:
10956:
10951:
10949:
10934:
10932:
10931:
10926:
10914:
10912:
10911:
10906:
10889:
10874:
10872:
10871:
10866:
10850:
10848:
10847:
10842:
10830:
10828:
10827:
10822:
10820:
10819:
10806:
10804:
10803:
10798:
10786:
10784:
10783:
10778:
10766:
10764:
10763:
10758:
10756:
10755:
10739:. The latter is
10738:
10736:
10735:
10730:
10725:
10724:
10709:
10694:
10692:
10691:
10686:
10667:
10665:
10664:
10659:
10654:
10653:
10648:
10647:
10631:
10616:
10614:
10613:
10608:
10596:
10594:
10593:
10588:
10571:
10569:
10568:
10563:
10561:
10560:
10551:
10550:
10549:
10548:
10534:
10533:
10524:
10523:
10518:
10517:
10497:quotient modules
10494:
10492:
10491:
10486:
10474:
10472:
10471:
10466:
10454:
10452:
10451:
10446:
10434:
10432:
10431:
10426:
10410:
10408:
10407:
10402:
10400:
10399:
10367:In the realm of
10358:
10336:
10334:
10333:
10328:
10323:
10319:
10280:
10257:
10256:
10251:
10236:
10218:
10216:
10215:
10210:
10208:
10207:
10202:
10193:
10192:
10187:
10178:
10177:
10161:
10159:
10158:
10153:
10151:
10150:
10145:
10123:
10121:
10120:
10115:
10110:
10106:
10102:
10101:
10099:
10094:
10077:
10071:
10066:
10045:
10044:
10043:
10042:
10032:
10020:
10019:
9999:
9958:
9940:
9938:
9937:
9932:
9902:
9900:
9899:
9894:
9882:
9873:
9871:
9870:
9865:
9860:
9856:
9817:
9794:
9793:
9788:
9773:
9755:
9753:
9752:
9747:
9735:
9733:
9732:
9727:
9725:
9724:
9719:
9710:
9709:
9704:
9695:
9694:
9678:
9676:
9675:
9670:
9668:
9667:
9662:
9639:
9637:
9636:
9631:
9629:
9625:
9621:
9620:
9619:
9618:
9608:
9596:
9595:
9574:
9573:
9568:
9542:
9501:
9483:
9481:
9480:
9475:
9463:
9461:
9460:
9455:
9453:
9452:
9436:
9434:
9433:
9428:
9425:
9420:
9400:
9398:
9397:
9392:
9387:
9386:
9384:
9379:
9369:
9356:
9351:
9330:
9329:
9328:
9327:
9317:
9305:
9304:
9279:
9277:
9276:
9271:
9263:
9262:
9246:
9244:
9243:
9238:
9226:
9224:
9223:
9218:
9201:
9186:
9184:
9183:
9178:
9166:
9164:
9163:
9158:
9131:orthogonal group
9128:
9126:
9125:
9120:
9108:
9106:
9105:
9100:
9088:
9086:
9085:
9080:
9078:
9066:
9064:
9063:
9058:
9056:
9044:
9042:
9041:
9036:
9034:
9019:
9017:
9016:
9011:
9000:
8981:
8979:
8978:
8973:
8942:
8940:
8939:
8934:
8923:
8908:
8906:
8905:
8900:
8877:
8875:
8874:
8869:
8858:
8843:
8841:
8840:
8835:
8833:
8821:
8819:
8818:
8813:
8811:
8796:
8794:
8793:
8788:
8769:
8767:
8766:
8761:
8756:
8742:
8716:
8698:
8696:
8695:
8690:
8673:
8655:
8653:
8652:
8647:
8636:
8618:
8616:
8615:
8610:
8599:
8585:
8584:
8569:
8568:
8552:
8537:
8514:
8513:
8501:
8471:
8469:
8468:
8463:
8461:
8460:
8440:
8438:
8437:
8432:
8408:
8406:
8405:
8400:
8388:
8386:
8385:
8380:
8369:
8345:
8343:
8342:
8337:
8332:
8331:
8300:
8298:
8297:
8292:
8275:
8254:
8252:
8251:
8246:
8234:
8232:
8231:
8226:
8218:
8217:
8201:
8199:
8198:
8193:
8181:
8179:
8178:
8173:
8155:
8153:
8152:
8147:
8136:
8107:
8105:
8104:
8099:
8093:
8088:
8079:
8078:
8068:
8063:
8045:
8044:
8023:
8021:
8020:
8015:
8010:
8009:
7991:
7990:
7971:
7969:
7968:
7963:
7961:
7960:
7955:
7936:
7934:
7933:
7928:
7916:
7914:
7913:
7908:
7906:
7905:
7889:
7887:
7886:
7881:
7876:
7875:
7857:
7856:
7837:
7835:
7834:
7829:
7802:self-adjointness
7795:
7793:
7792:
7787:
7785:
7784:
7768:
7751:
7749:
7748:
7743:
7729:
7727:
7726:
7721:
7713:
7712:
7692:
7690:
7689:
7684:
7670:
7668:
7667:
7662:
7654:
7653:
7633:
7631:
7630:
7625:
7610:
7608:
7607:
7602:
7597:
7567:
7565:
7564:
7559:
7541:
7539:
7538:
7533:
7528:
7505:
7475:
7473:
7472:
7467:
7462:
7461:
7456:
7441:
7423:
7421:
7420:
7415:
7403:
7401:
7400:
7395:
7393:
7392:
7387:
7368:
7366:
7365:
7360:
7355:
7354:
7336:
7335:
7311:
7309:
7308:
7303:
7297:
7292:
7283:
7282:
7272:
7267:
7249:
7248:
7227:
7225:
7224:
7219:
7214:
7213:
7208:
7193:
7172:
7170:
7169:
7164:
7162:
7161:
7156:
7143:
7141:
7140:
7135:
7123:
7121:
7120:
7115:
7110:
7074:
7072:
7071:
7066:
7064:
7063:
7058:
7045:
7043:
7042:
7037:
7025:
7023:
7022:
7017:
7012:
7011:
7006:
6991:
6976:
6974:
6973:
6968:
6963:
6929:
6927:
6926:
6921:
6891:
6889:
6888:
6883:
6869:
6867:
6866:
6861:
6853:
6852:
6832:
6830:
6829:
6824:
6810:
6808:
6807:
6802:
6794:
6793:
6773:
6771:
6770:
6765:
6750:
6748:
6747:
6742:
6737:
6707:
6705:
6704:
6699:
6694:
6671:
6641:
6639:
6638:
6633:
6615:
6613:
6612:
6607:
6602:
6543:
6541:
6540:
6535:
6519:
6517:
6511:
6509:
6508:
6503:
6491:
6489:
6488:
6483:
6475:
6449:
6447:
6446:
6441:
6433:
6432:
6431:
6415:
6414:
6396:
6360:
6358:
6357:
6352:
6335:
6313:
6311:
6310:
6305:
6291:
6290:
6275:
6255:
6226:, respectively.
6225:
6223:
6222:
6217:
6215:
6214:
6203:
6190:
6188:
6187:
6182:
6180:
6179:
6168:
6155:
6153:
6152:
6147:
6129:
6127:
6126:
6121:
6116:
6115:
6110:
6095:
6080:
6078:
6077:
6072:
6067:
6066:
6061:
6046:
6031:
6029:
6028:
6023:
6021:
6020:
6015:
5996:
5994:
5993:
5988:
5986:
5985:
5980:
5956:
5954:
5953:
5948:
5943:
5939:
5923:
5911:
5910:
5898:
5897:
5870:
5830:
5810:
5808:
5807:
5802:
5790:
5788:
5787:
5782:
5770:
5768:
5767:
5762:
5745:
5726:
5724:
5723:
5718:
5707:
5696:
5695:
5679:
5677:
5676:
5671:
5659:
5657:
5656:
5651:
5634:
5632:
5631:
5626:
5624:
5623:
5614:
5613:
5598:
5595:
5576:
5573:
5547:
5546:
5525:
5523:
5522:
5517:
5515:
5514:
5498:
5496:
5495:
5490:
5473:
5471:
5470:
5465:
5463:
5462:
5429:
5427:
5426:
5421:
5409:
5407:
5406:
5401:
5379:
5377:
5376:
5371:
5357:
5356:
5340:
5338:
5337:
5332:
5320:
5318:
5317:
5312:
5300:
5298:
5297:
5292:
5280:
5278:
5277:
5272:
5260:
5258:
5257:
5252:
5209:
5207:
5206:
5201:
5190:
5189:
5184:
5168:
5166:
5165:
5160:
5155:
5154:
5153:
5152:
5134:
5133:
5116:
5115:
5114:
5113:
5095:
5094:
5070:
5068:
5067:
5062:
5060:
5059:
5058:
5057:
5039:
5038:
5017:
5015:
5014:
5009:
5007:
5006:
5005:
5004:
4986:
4985:
4975:
4974:
4966:
4955:
4953:
4952:
4947:
4945:
4944:
4928:
4926:
4925:
4920:
4908:
4906:
4905:
4900:
4882:
4880:
4879:
4874:
4871:
4870:
4869:
4851:
4850:
4840:
4839:
4838:
4820:
4819:
4809:
4808:
4800:
4784:
4782:
4781:
4776:
4771:
4770:
4757:
4756:
4755:
4737:
4736:
4726:
4725:
4724:
4706:
4705:
4695:
4694:
4686:
4679:
4678:
4677:
4676:
4658:
4657:
4647:
4646:
4638:
4631:
4630:
4629:
4628:
4610:
4609:
4599:
4598:
4590:
4574:
4572:
4571:
4566:
4564:
4563:
4562:
4561:
4543:
4542:
4521:
4519:
4518:
4513:
4511:
4510:
4509:
4508:
4490:
4489:
4463:
4461:
4460:
4455:
4450:
4449:
4448:
4447:
4429:
4428:
4414:
4413:
4412:
4411:
4393:
4392:
4382:
4381:
4373:
4366:
4365:
4364:
4363:
4345:
4344:
4330:
4329:
4328:
4327:
4309:
4308:
4298:
4297:
4289:
4273:
4271:
4270:
4265:
4263:
4262:
4261:
4260:
4242:
4241:
4220:
4218:
4217:
4212:
4210:
4209:
4208:
4207:
4189:
4188:
4167:
4165:
4164:
4159:
4157:
4156:
4155:
4154:
4136:
4135:
4118:
4117:
4116:
4115:
4097:
4096:
4075:
4073:
4072:
4067:
4065:
4064:
4063:
4062:
4044:
4043:
4022:
4020:
4019:
4014:
4009:
4008:
3990:
3989:
3970:
3968:
3967:
3962:
3939:
3937:
3936:
3931:
3929:
3928:
3916:
3915:
3914:
3913:
3895:
3894:
3871:
3870:
3869:
3868:
3850:
3849:
3839:
3838:
3830:
3814:
3812:
3811:
3806:
3776:
3774:
3773:
3768:
3757:to the space of
3756:
3754:
3753:
3748:
3746:
3745:
3744:
3743:
3725:
3724:
3699:
3697:
3696:
3691:
3689:
3688:
3687:
3686:
3668:
3667:
3646:
3644:
3643:
3638:
3636:
3635:
3634:
3633:
3615:
3614:
3593:
3589:
3587:
3586:
3581:
3579:
3578:
3577:
3576:
3558:
3557:
3536:
3534:
3533:
3528:
3517:
3516:
3511:
3489:
3487:
3486:
3483:{\displaystyle }
3481:
3463:
3461:
3460:
3455:
3443:
3441:
3440:
3435:
3417:
3415:
3414:
3409:
3404:
3403:
3385:
3384:
3365:
3363:
3362:
3357:
3354:
3346:
3345:
3344:
3326:
3325:
3301:
3299:
3298:
3293:
3291:
3290:
3289:
3288:
3270:
3269:
3248:
3246:
3245:
3240:
3210:
3208:
3207:
3202:
3200:
3199:
3198:
3197:
3179:
3178:
3157:
3155:
3154:
3149:
3137:
3135:
3134:
3129:
3127:
3126:
3110:
3108:
3107:
3102:
3090:
3088:
3087:
3082:
3080:
3079:
3078:
3077:
3059:
3058:
3037:
3035:
3034:
3029:
3011:
3009:
3008:
3003:
2991:
2989:
2988:
2983:
2972:
2971:
2966:
2944:
2942:
2941:
2936:
2934:
2933:
2932:
2931:
2913:
2912:
2891:
2889:
2888:
2883:
2875:
2874:
2856:
2855:
2830:
2828:
2827:
2822:
2802:
2800:
2799:
2794:
2782:
2780:
2779:
2774:
2769:
2768:
2750:
2749:
2730:
2728:
2727:
2722:
2720:
2719:
2718:
2717:
2699:
2698:
2677:
2675:
2674:
2669:
2651:
2649:
2648:
2643:
2635:
2634:
2616:
2615:
2593:
2591:
2590:
2585:
2573:
2571:
2570:
2565:
2553:
2551:
2550:
2545:
2533:
2531:
2530:
2525:
2513:
2511:
2510:
2505:
2500:
2499:
2477:
2475:
2474:
2469:
2457:
2455:
2454:
2449:
2414:
2412:
2411:
2406:
2404:
2403:
2396:
2395:
2362:
2361:
2332:
2331:
2322:
2321:
2294:
2293:
2270:
2269:
2268:
2260:
2259:
2251:
2224:
2222:
2221:
2216:
2204:
2202:
2201:
2196:
2185:. If the first
2181:) to obtain its
2180:
2178:
2177:
2172:
2136:
2134:
2133:
2128:
2112:
2110:
2109:
2104:
2092:
2090:
2089:
2084:
2063:
2061:
2060:
2057:{\displaystyle }
2055:
2037:
2035:
2034:
2029:
2017:
2015:
2014:
2009:
1997:
1995:
1994:
1989:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1918:
1916:
1915:
1910:
1869:
1867:
1866:
1861:
1850:
1849:
1841:
1827:if and only if
1826:
1824:
1823:
1818:
1807:
1806:
1801:
1779:
1777:
1776:
1771:
1769:
1768:
1760:
1750:
1748:
1747:
1742:
1730:
1728:
1727:
1722:
1692:
1690:
1689:
1684:
1682:
1681:
1665:
1663:
1662:
1657:
1645:
1643:
1642:
1637:
1618:
1616:
1615:
1610:
1592:
1590:
1589:
1584:
1573:
1572:
1567:
1541:
1539:
1538:
1533:
1528:
1527:
1509:
1508:
1486:
1484:
1483:
1478:
1466:
1464:
1463:
1458:
1447:
1446:
1441:
1425:
1423:
1422:
1417:
1399:
1397:
1396:
1391:
1379:
1377:
1376:
1371:
1366:
1365:
1347:
1346:
1327:
1325:
1324:
1319:
1317:
1316:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1256:
1254:
1253:
1248:
1237:
1236:
1231:
1207:
1196:projective plane
1193:
1187:
1180:
1169:
1159:
1152:
1150:
1149:
1144:
1142:
1141:
1130:
1117:projective space
1114:
1110:
1099:
1068:
1066:
1065:
1060:
1048:
1046:
1045:
1040:
1029:over a manifold
1017:
1015:
1014:
1009:
997:
995:
994:
989:
977:
975:
974:
969:
951:
949:
948:
943:
938:
937:
932:
923:
922:
917:
901:
897:
895:
894:
889:
873:
871:
870:
865:
863:
862:
857:
844:
842:
841:
836:
834:
833:
828:
815:
813:
812:
807:
791:
789:
788:
783:
766:. At each point
765:
763:
762:
757:
755:
754:
749:
736:
734:
733:
728:
716:
714:
713:
708:
657:
655:
654:
649:
637:
635:
634:
629:
617:
615:
614:
609:
597:
595:
594:
589:
572:
557:
555:
554:
549:
538:
537:
532:
516:
514:
513:
508:
491:
476:
474:
473:
468:
457:
456:
451:
420:
418:
417:
412:
407:
406:
401:
392:
391:
386:
359:
357:
356:
351:
323:smooth manifolds
309:
307:
306:
301:
289:
287:
286:
281:
269:
267:
266:
261:
250:
238:projective space
235:
233:
232:
227:
215:
213:
212:
207:
196:
195:
190:
174:
172:
171:
166:
151:
149:
148:
143:
128:
126:
125:
120:
107:linear subspaces
101:
99:
98:
93:
73:
71:
70:
65:
54:
53:
48:
21361:
21360:
21356:
21355:
21354:
21352:
21351:
21350:
21321:
21320:
21314:
21278:
21262:Mattila, Pertti
21246:
21236:Springer-Verlag
21217:
21198:
21176:
21158:Milnor, John W.
21151:(2.0 ed.).
21148:
21116:
21108:. p. 211.
21089:
21084:
21074:
21072:
21027:
21019:Morel, Fabien;
21017:
21013:
20988:
20984:
20935:
20931:
20921:
20919:
20917:Quanta Magazine
20909:
20905:
20856:
20852:
20811:
20807:
20792:
20767:
20763:
20736:
20732:
20717:
20692:
20685:
20648:
20641:
20602:
20595:
20591:, 30–46 (1981).
20586:
20582:
20539:
20535:
20520:10.2307/2373211
20504:
20500:
20467:10.2307/2372896
20451:
20447:
20410:
20406:
20375:
20371:
20356:
20336:
20329:
20312:
20308:
20286:
20278:, p. 211,
20261:
20257:
20249:
20245:
20237:
20227:Springer-Verlag
20214:
20210:
20202:
20198:
20190:
20186:
20178:
20174:
20170:
20162:Grassmann graph
20058:
20046:computer vision
20009:KP Tau function
19971:
19963:Cartan quadrics
19873:
19870:
19869:
19842:
19839:
19838:
19822:
19819:
19818:
19792:
19784:
19783:
19759:
19754:
19746:
19743:
19740:
19739:
19707:
19702:
19694:
19691:
19688:
19687:
19671:
19668:
19667:
19651:
19648:
19647:
19631:
19628:
19627:
19624:
19555:
19528:
19523:
19522:
19513:
19499:
19497:
19496:
19495:
19493:
19490:
19489:
19463:
19458:
19457:
19448:
19434:
19432:
19431:
19430:
19428:
19425:
19424:
19404:
19399:
19398:
19389:
19381:
19380:
19378:
19375:
19374:
19357:
19352:
19351:
19349:
19346:
19345:
19329:
19326:
19325:
19319:
19287:
19283:
19281:
19278:
19277:
19242:
19234:
19233:
19218:
19214:
19212:
19209:
19208:
19167:
19163:
19161:
19158:
19157:
19126:
19123:
19122:
19085:
19081:
19054:
19050:
19048:
19045:
19044:
19014:
19011:
19010:
18943:
18939:
18915:
18911:
18909:
18906:
18905:
18874:
18866:
18865:
18857:
18854:
18853:
18827:
18819:
18818:
18810:
18807:
18806:
18781:
18778:
18777:
18753:
18749:
18747:
18744:
18743:
18717:
18709:
18708:
18706:
18703:
18702:
18686:
18683:
18682:
18666:
18663:
18662:
18659:
18635:
18632:
18631:
18608:
18605:
18604:
18568:
18564:
18531:
18527:
18512:
18508:
18506:
18503:
18502:
18441:
18438:
18437:
18408:
18405:
18404:
18388:
18385:
18384:
18383:of the bundles
18364:
18361:
18360:
18344:
18341:
18340:
18321:
18318:
18317:
18289:
18286:
18285:
18257:
18254:
18253:
18229:
18226:
18225:
18206:
18203:
18202:
18186:
18183:
18182:
18162:
18157:
18156:
18147:
18139:
18138:
18136:
18133:
18132:
18129:
18101:
18100:
18086:
18082:
18079:
18078:
18064:
18060:
18053:
18052:
18037:
18033:
18031:
18028:
18027:
18002:
17999:
17998:
17978:
17975:
17974:
17945:
17941:
17939:
17936:
17935:
17897:
17893:
17878:
17874:
17852:
17848:
17842:
17838:
17802:
17798:
17783:
17779:
17777:
17774:
17773:
17742:
17737:
17736:
17727:
17719:
17718:
17716:
17713:
17712:
17696:
17693:
17692:
17666:
17661:
17660:
17645:
17637:
17636:
17634:
17631:
17630:
17624:
17618:
17614:
17594:
17589:
17588:
17579:
17571:
17570:
17568:
17565:
17564:
17547:
17542:
17541:
17533:
17531:
17528:
17527:
17511:
17508:
17507:
17501:
17497:
17477:
17472:
17471:
17462:
17454:
17453:
17451:
17448:
17447:
17424:
17420:
17418:
17415:
17414:
17403:) are known as
17361:
17357:
17338:
17334:
17313:
17297:
17289:
17288:
17255:
17251:
17249:
17246:
17245:
17220:
17216:
17211:
17208:
17207:
17177:
17169:
17168:
17160:
17157:
17156:
17130:
17122:
17121:
17097:
17093:
17091:
17088:
17087:
17070:
17066:
17052:
17049:
17048:
17015:
17011:
16996:
16992:
16990:
16987:
16986:
16961:
16957:
16951:
16940:
16928:
16920:
16918:
16915:
16914:
16886:
16882:
16867:
16863:
16852:
16849:
16848:
16816:
16812:
16798:
16796:
16793:
16792:
16769:
16765:
16750:
16746:
16737:
16733:
16731:
16728:
16727:
16697:
16689:
16688:
16686:
16683:
16682:
16669:affine subpaces
16665:
16628:
16625:
16624:
16608:
16605:
16604:
16587:
16583:
16581:
16578:
16577:
16561:
16558:
16557:
16525:
16517:
16516:
16514:
16511:
16510:
16484:
16476:
16475:
16473:
16470:
16469:
16452:
16448:
16446:
16443:
16442:
16426:
16423:
16422:
16405:
16401:
16392:
16388:
16386:
16383:
16382:
16360:
16357:
16356:
16331:
16327:
16310:
16302:
16284:
16276:
16275:
16273:
16270:
16269:
16247:
16243:
16241:
16238:
16237:
16209:
16206:
16205:
16189:
16186:
16185:
16169:
16166:
16165:
16136:
16132:
16123:
16119:
16110:
16106:
16098:
16084:
16081:
16080:
16059:
16055:
16053:
16050:
16049:
16032:
16028:
16026:
16023:
16022:
16005:
16001:
15993:
15985:
15982:
15981:
15943:
15920:
15917:
15916:
15891:
15888:
15887:
15868:
15863:
15860:
15859:
15828:
15825:
15824:
15802:
15799:
15798:
15782:
15779:
15778:
15775:
15747:
15743:
15735:
15733:
15730:
15729:
15700:
15692:
15691:
15683:
15680:
15679:
15651:
15647:
15641:
15637:
15628:
15624:
15618:
15614:
15605:
15601:
15595:
15591:
15589:
15586:
15585:
15560:
15556:
15547:
15543:
15534:
15530:
15521:
15517:
15508:
15504:
15495:
15491:
15486:
15483:
15482:
15459:
15455:
15447:
15429:
15421:
15420:
15412:
15409:
15408:
15386:
15383:
15382:
15348:
15345:
15344:
15321:
15317:
15315:
15312:
15311:
15286:
15283:
15282:
15261:
15257:
15255:
15252:
15251:
15228:
15224:
15209:
15205:
15203:
15200:
15199:
15176:
15172:
15154:
15150:
15148:
15147:
15132:
15128:
15126:
15123:
15122:
15083:
15079:
15061:
15057:
15055:
15054:
15039:
15035:
15034:
15030:
15022:
15018:
15003:
14999:
14984:
14980:
14979:
14975:
14969:
14965:
14944:
14933:
14927:
14924:
14923:
14892:
14884:
14883:
14881:
14878:
14877:
14848:
14840:
14839:
14831:
14828:
14827:
14797:
14794:
14793:
14771:
14768:
14767:
14738:
14734:
14722:
14718:
14709:
14705:
14697:
14694:
14693:
14664:
14660:
14648:
14644:
14635:
14631:
14623:
14620:
14619:
14593:
14589:
14581:
14579:
14576:
14575:
14558:
14555:
14554:
14538:
14535:
14534:
14514:
14510:
14495:
14491:
14486:
14483:
14482:
14462:
14458:
14456:
14453:
14452:
14436:
14433:
14432:
14407:
14404:
14403:
14387:
14384:
14383:
14363:
14359:
14344:
14340:
14335:
14332:
14331:
14305:
14297:
14296:
14288:
14285:
14284:
14283:of the element
14279:are called the
14254:
14250:
14235:
14231:
14212:
14208:
14193:
14189:
14188:
14184:
14179:
14176:
14175:
14174:. The elements
14153:
14149:
14148:
14144:
14127:
14123:
14122:
14118:
14113:
14110:
14109:
14087:
14084:
14083:
14064:
14060:
14045:
14041:
14040:
14036:
14034:
14031:
14030:
14014:
14011:
14010:
13987:
13983:
13968:
13964:
13956:
13953:
13952:
13926:
13925:
13916:
13912:
13910:
13905:
13896:
13892:
13889:
13888:
13883:
13878:
13872:
13871:
13862:
13858:
13856:
13851:
13845:
13841:
13834:
13833:
13821:
13817:
13807:
13803:
13791:
13787:
13785:
13782:
13781:
13760:
13757:
13756:
13736:
13732:
13717:
13713:
13708:
13705:
13704:
13688:
13685:
13684:
13667:
13663:
13661:
13658:
13657:
13634:
13630:
13612:
13608:
13603:
13600:
13599:
13579:
13575:
13560:
13556:
13551:
13548:
13547:
13531:
13528:
13527:
13511:
13508:
13507:
13485:
13482:
13481:
13465:
13462:
13461:
13441:
13437:
13422:
13418:
13413:
13410:
13409:
13393:
13390:
13389:
13373:
13370:
13369:
13346:
13342:
13334:
13332:
13329:
13328:
13302:
13294:
13293:
13291:
13288:
13287:
13274:
13229:
13226:
13225:
13198:
13195:
13194:
13165:
13162:
13161:
13145:
13142:
13141:
13125:
13122:
13121:
13118:change of basis
13097:
13094:
13093:
13066:
13063:
13062:
13039:
13035:
13020:
13016:
12996:
12993:
12992:
12967:
12964:
12963:
12947:
12944:
12943:
12923:
12919:
12904:
12900:
12895:
12892:
12891:
12866:
12863:
12862:
12846:
12843:
12842:
12826:
12823:
12822:
12806:
12803:
12802:
12780:
12777:
12776:
12744:
12740:
12739:
12735:
12730:
12704:
12697:
12694:
12693:
12672:
12669:
12668:
12648:
12644:
12642:
12639:
12638:
12619:
12616:
12615:
12581:
12579:
12576:
12575:
12568:
12562:
12506:
12505:
12488:
12462:
12443:
12439:
12437:
12434:
12433:
12404:
12403:
12386:
12380:
12376:
12362:
12361:
12358:
12357:
12323:
12322:
12314:
12312:
12309:
12308:
12283:
12281:
12278:
12277:
12255:
12252:
12251:
12222:
12221:
12204:
12202:
12199:
12198:
12182:
12179:
12178:
12161:
12160:
12158:
12155:
12154:
12138:
12135:
12134:
12112:
12109:
12108:
12083:
12079:
12070:
12069:
12061:
12059:
12056:
12055:
12039:
12036:
12035:
12006:
12005:
11988:
11986:
11983:
11982:
11966:
11963:
11962:
11945:
11939:
11938:
11937:
11935:
11932:
11931:
11914:
11910:
11908:
11905:
11904:
11872:
11868:
11859:
11858:
11850:
11838:
11832:
11831:
11830:
11822:
11820:
11817:
11816:
11783:
11782:
11765:
11757:
11754:
11753:
11744:
11716:
11715:
11698:
11692:
11688:
11679:
11678:
11670:
11652:
11651:
11634:
11633:
11627:
11626:
11625:
11621:
11616:
11604:
11603:
11595:
11593:
11590:
11589:
11579:. The quotient
11560:
11559:
11542:
11540:
11537:
11536:
11520:
11517:
11516:
11494:
11493:
11476:
11475:
11469:
11468:
11467:
11465:
11462:
11461:
11444:
11443:
11441:
11438:
11437:
11407:
11406:
11389:
11388:
11382:
11381:
11380:
11373:
11369:
11361:
11359:
11356:
11355:
11338:
11337:
11335:
11332:
11331:
11328:
11298:
11295:
11294:
11257:
11253:
11252:
11248:
11242:
11241:
11224:
11222:
11219:
11218:
11201:
11197:
11191:
11190:
11173:
11171:
11168:
11167:
11133:
11119:
11116:
11115:
11099:
11096:
11095:
11079:
11076:
11075:
11046:
11045:
11039:
11038:
11037:
11020:
11009:
11003:
10999:
10990:
10989:
10972:
10970:
10967:
10966:
10942:
10940:
10937:
10936:
10920:
10917:
10916:
10882:
10880:
10877:
10876:
10860:
10857:
10856:
10836:
10833:
10832:
10815:
10814:
10812:
10809:
10808:
10792:
10789:
10788:
10772:
10769:
10768:
10751:
10750:
10748:
10745:
10744:
10720:
10719:
10702:
10700:
10697:
10696:
10680:
10677:
10676:
10649:
10643:
10642:
10641:
10624:
10622:
10619:
10618:
10602:
10599:
10598:
10582:
10579:
10578:
10556:
10552:
10544:
10540:
10539:
10535:
10529:
10528:
10519:
10513:
10512:
10511:
10509:
10506:
10505:
10480:
10477:
10476:
10460:
10457:
10456:
10455:. Then to each
10440:
10437:
10436:
10420:
10417:
10416:
10395:
10394:
10392:
10389:
10388:
10385:
10365:
10345:
10285:
10281:
10276:
10252:
10247:
10246:
10229:
10227:
10224:
10223:
10203:
10198:
10197:
10188:
10183:
10182:
10173:
10169:
10167:
10164:
10163:
10146:
10141:
10140:
10132:
10129:
10128:
10095:
10090:
10085:
10081:
10073:
10067:
10062:
10038:
10034:
10033:
10028:
10027:
10015:
10011:
10004:
10000:
9995:
9951:
9949:
9946:
9945:
9911:
9908:
9907:
9888:
9885:
9884:
9878:
9822:
9818:
9813:
9789:
9784:
9783:
9766:
9764:
9761:
9760:
9741:
9738:
9737:
9720:
9715:
9714:
9705:
9700:
9699:
9690:
9686:
9684:
9681:
9680:
9663:
9658:
9657:
9649:
9646:
9645:
9614:
9610:
9609:
9604:
9603:
9591:
9587:
9569:
9564:
9563:
9547:
9543:
9538:
9494:
9492:
9489:
9488:
9469:
9466:
9465:
9448:
9444:
9442:
9439:
9438:
9421:
9416:
9410:
9407:
9406:
9380:
9375:
9370:
9365:
9364:
9352:
9347:
9323:
9319:
9318:
9313:
9312:
9300:
9296:
9288:
9285:
9284:
9258:
9254:
9252:
9249:
9248:
9232:
9229:
9228:
9194:
9192:
9189:
9188:
9172:
9169:
9168:
9137:
9134:
9133:
9114:
9111:
9110:
9094:
9091:
9090:
9074:
9072:
9069:
9068:
9052:
9050:
9047:
9046:
9030:
9028:
9025:
9024:
8993:
8991:
8988:
8987:
8967:
8964:
8963:
8945:algebraic group
8916:
8914:
8911:
8910:
8894:
8891:
8890:
8884:smooth manifold
8851:
8849:
8846:
8845:
8829:
8827:
8824:
8823:
8807:
8805:
8802:
8801:
8782:
8779:
8778:
8752:
8735:
8709:
8707:
8704:
8703:
8666:
8664:
8661:
8660:
8629:
8627:
8624:
8623:
8592:
8580:
8576:
8564:
8560:
8548:
8530:
8509:
8505:
8488:
8480:
8477:
8476:
8456:
8452:
8450:
8447:
8446:
8414:
8411:
8410:
8394:
8391:
8390:
8362:
8354:
8351:
8350:
8327:
8323:
8309:
8306:
8305:
8268:
8260:
8257:
8256:
8240:
8237:
8236:
8213:
8209:
8207:
8204:
8203:
8187:
8184:
8183:
8167:
8164:
8163:
8129:
8127:
8124:
8123:
8116:
8089:
8084:
8074:
8070:
8064:
8053:
8040:
8036:
8034:
8031:
8030:
8005:
8001:
7986:
7982:
7977:
7974:
7973:
7956:
7951:
7950:
7942:
7939:
7938:
7922:
7919:
7918:
7901:
7897:
7895:
7892:
7891:
7871:
7867:
7852:
7848:
7843:
7840:
7839:
7809:
7806:
7805:
7761:
7760:
7759:
7757:
7754:
7753:
7737:
7734:
7733:
7708:
7704:
7702:
7699:
7698:
7678:
7675:
7674:
7649:
7645:
7643:
7640:
7639:
7619:
7616:
7615:
7593:
7573:
7570:
7569:
7547:
7544:
7543:
7524:
7501:
7481:
7478:
7477:
7457:
7452:
7451:
7434:
7432:
7429:
7428:
7409:
7406:
7405:
7388:
7383:
7382:
7374:
7371:
7370:
7350:
7346:
7331:
7327:
7322:
7319:
7318:
7293:
7288:
7278:
7274:
7268:
7257:
7244:
7240:
7238:
7235:
7234:
7209:
7204:
7203:
7186:
7178:
7175:
7174:
7157:
7152:
7151:
7149:
7146:
7145:
7129:
7126:
7125:
7106:
7080:
7077:
7076:
7059:
7054:
7053:
7051:
7048:
7047:
7031:
7028:
7027:
7007:
7002:
7001:
6984:
6982:
6979:
6978:
6959:
6939:
6936:
6935:
6897:
6894:
6893:
6877:
6874:
6873:
6848:
6844:
6842:
6839:
6838:
6818:
6815:
6814:
6789:
6785:
6783:
6780:
6779:
6759:
6756:
6755:
6733:
6713:
6710:
6709:
6690:
6667:
6647:
6644:
6643:
6621:
6618:
6617:
6598:
6584:
6581:
6580:
6577:
6529:
6526:
6525:
6515:
6513:
6497:
6494:
6493:
6468:
6460:
6457:
6456:
6424:
6423:
6419:
6410:
6406:
6389:
6375:
6372:
6371:
6328:
6326:
6323:
6322:
6286:
6282:
6268:
6245:
6234:
6231:
6230:
6204:
6199:
6198:
6196:
6193:
6192:
6169:
6164:
6163:
6161:
6158:
6157:
6135:
6132:
6131:
6111:
6106:
6105:
6088:
6086:
6083:
6082:
6062:
6057:
6056:
6039:
6037:
6034:
6033:
6016:
6011:
6010:
6002:
5999:
5998:
5981:
5976:
5975:
5967:
5964:
5963:
5916:
5906:
5902:
5893:
5889:
5860:
5853:
5849:
5823:
5821:
5818:
5817:
5796:
5793:
5792:
5776:
5773:
5772:
5738:
5736:
5733:
5732:
5700:
5691:
5687:
5685:
5682:
5681:
5665:
5662:
5661:
5645:
5642:
5641:
5619:
5618:
5609:
5605:
5594:
5587:
5586:
5572:
5561:
5560:
5542:
5538:
5536:
5533:
5532:
5510:
5506:
5504:
5501:
5500:
5484:
5481:
5480:
5458:
5454:
5440:
5437:
5436:
5415:
5412:
5411:
5389:
5386:
5385:
5352:
5348:
5346:
5343:
5342:
5326:
5323:
5322:
5306:
5303:
5302:
5286:
5283:
5282:
5266:
5263:
5262:
5234:
5231:
5230:
5216:
5185:
5177:
5176:
5174:
5171:
5170:
5148:
5144:
5129:
5125:
5124:
5120:
5109:
5105:
5090:
5086:
5085:
5081:
5076:
5073:
5072:
5053:
5049:
5034:
5030:
5029:
5025:
5023:
5020:
5019:
5000:
4996:
4981:
4977:
4976:
4965:
4964:
4963:
4961:
4958:
4957:
4940:
4936:
4934:
4931:
4930:
4914:
4911:
4910:
4888:
4885:
4884:
4865:
4861:
4846:
4842:
4841:
4834:
4830:
4815:
4811:
4810:
4799:
4798:
4795:
4792:
4791:
4763:
4759:
4751:
4747:
4732:
4728:
4727:
4720:
4716:
4701:
4697:
4696:
4685:
4684:
4672:
4668:
4653:
4649:
4648:
4637:
4636:
4635:
4624:
4620:
4605:
4601:
4600:
4589:
4588:
4587:
4585:
4582:
4581:
4557:
4553:
4538:
4534:
4533:
4529:
4527:
4524:
4523:
4504:
4500:
4485:
4481:
4480:
4476:
4474:
4471:
4470:
4443:
4439:
4424:
4420:
4419:
4415:
4407:
4403:
4388:
4384:
4383:
4372:
4371:
4370:
4359:
4355:
4340:
4336:
4335:
4331:
4323:
4319:
4304:
4300:
4299:
4288:
4287:
4286:
4284:
4281:
4280:
4256:
4252:
4237:
4233:
4232:
4228:
4226:
4223:
4222:
4203:
4199:
4184:
4180:
4179:
4175:
4173:
4170:
4169:
4150:
4146:
4131:
4127:
4126:
4122:
4111:
4107:
4092:
4088:
4087:
4083:
4081:
4078:
4077:
4058:
4054:
4039:
4035:
4034:
4030:
4028:
4025:
4024:
4004:
4000:
3985:
3981:
3976:
3973:
3972:
3950:
3947:
3946:
3921:
3917:
3909:
3905:
3890:
3886:
3885:
3881:
3864:
3860:
3845:
3841:
3840:
3829:
3828:
3827:
3825:
3822:
3821:
3782:
3779:
3778:
3762:
3759:
3758:
3739:
3735:
3720:
3716:
3715:
3711:
3709:
3706:
3705:
3682:
3678:
3663:
3659:
3658:
3654:
3652:
3649:
3648:
3629:
3625:
3610:
3606:
3605:
3601:
3599:
3596:
3595:
3591:
3572:
3568:
3553:
3549:
3548:
3544:
3542:
3539:
3538:
3512:
3504:
3503:
3495:
3492:
3491:
3469:
3466:
3465:
3449:
3446:
3445:
3423:
3420:
3419:
3399:
3395:
3380:
3376:
3371:
3368:
3367:
3347:
3340:
3336:
3321:
3317:
3316:
3307:
3304:
3303:
3284:
3280:
3265:
3261:
3260:
3256:
3254:
3251:
3250:
3216:
3213:
3212:
3193:
3189:
3174:
3170:
3169:
3165:
3163:
3160:
3159:
3143:
3140:
3139:
3122:
3118:
3116:
3113:
3112:
3111:-th row is the
3096:
3093:
3092:
3073:
3069:
3054:
3050:
3049:
3045:
3043:
3040:
3039:
3017:
3014:
3013:
2997:
2994:
2993:
2967:
2959:
2958:
2950:
2947:
2946:
2927:
2923:
2908:
2904:
2903:
2899:
2897:
2894:
2893:
2870:
2866:
2851:
2847:
2839:
2836:
2835:
2816:
2813:
2812:
2809:identity matrix
2788:
2785:
2784:
2764:
2760:
2745:
2741:
2736:
2733:
2732:
2713:
2709:
2694:
2690:
2689:
2685:
2683:
2680:
2679:
2657:
2654:
2653:
2630:
2626:
2611:
2607:
2599:
2596:
2595:
2579:
2576:
2575:
2559:
2556:
2555:
2539:
2536:
2535:
2519:
2516:
2515:
2492:
2488:
2483:
2480:
2479:
2463:
2460:
2459:
2425:
2422:
2421:
2398:
2397:
2379:
2375:
2373:
2368:
2363:
2345:
2341:
2338:
2337:
2330:
2324:
2323:
2311:
2307:
2305:
2300:
2295:
2283:
2279:
2276:
2275:
2266:
2265:
2257:
2256:
2249:
2248:
2238:
2237:
2235:
2232:
2231:
2210:
2207:
2206:
2190:
2187:
2186:
2142:
2139:
2138:
2122:
2119:
2118:
2113:, we can apply
2098:
2095:
2094:
2072:
2069:
2068:
2043:
2040:
2039:
2023:
2020:
2019:
2003:
2000:
1999:
1977:
1974:
1973:
1957:
1954:
1953:
1931:
1928:
1927:
1880:
1877:
1876:
1840:
1839:
1837:
1834:
1833:
1802:
1794:
1793:
1785:
1782:
1781:
1759:
1758:
1756:
1753:
1752:
1736:
1733:
1732:
1698:
1695:
1694:
1677:
1673:
1671:
1668:
1667:
1651:
1648:
1647:
1631:
1628:
1627:
1598:
1595:
1594:
1568:
1560:
1559:
1551:
1548:
1547:
1546:of the element
1523:
1519:
1504:
1500:
1495:
1492:
1491:
1490:column vectors
1472:
1469:
1468:
1442:
1434:
1433:
1431:
1428:
1427:
1405:
1402:
1401:
1385:
1382:
1381:
1361:
1357:
1342:
1338:
1333:
1330:
1329:
1312:
1308:
1306:
1303:
1302:
1286:
1283:
1282:
1266:
1263:
1262:
1232:
1224:
1223:
1221:
1218:
1217:
1214:
1202:
1189:
1182:
1175:
1164:
1154:
1131:
1126:
1125:
1123:
1120:
1119:
1112:
1101:
1094:
1091:
1054:
1051:
1050:
1034:
1031:
1030:
1003:
1000:
999:
983:
980:
979:
957:
954:
953:
933:
928:
927:
918:
910:
909:
907:
904:
903:
899:
883:
880:
879:
858:
853:
852:
850:
847:
846:
829:
824:
823:
821:
818:
817:
801:
798:
797:
771:
768:
767:
750:
745:
744:
742:
739:
738:
722:
719:
718:
702:
699:
698:
695:Euclidean space
691:tangent bundles
664:
643:
640:
639:
623:
620:
619:
603:
600:
599:
565:
563:
560:
559:
533:
525:
524:
522:
519:
518:
484:
482:
479:
478:
452:
444:
443:
441:
438:
437:
402:
397:
396:
387:
379:
378:
376:
373:
372:
330:
327:
326:
325:, of dimension
295:
292:
291:
275:
272:
271:
246:
244:
241:
240:
221:
218:
217:
191:
183:
182:
180:
177:
176:
160:
157:
156:
137:
134:
133:
114:
111:
110:
87:
84:
83:
49:
41:
40:
38:
35:
34:
24:
17:
12:
11:
5:
21359:
21349:
21348:
21343:
21338:
21333:
21319:
21318:
21312:
21282:
21276:
21258:
21244:
21220:
21215:
21202:
21196:
21181:
21174:
21154:
21140:Hatcher, Allen
21136:
21114:
21098:Harris, Joseph
21088:
21085:
21083:
21082:
21040:(90): 45–143.
21011:
20982:
20929:
20903:
20866:(3): 637–699.
20850:
20805:
20790:
20761:
20730:
20715:
20683:
20639:
20593:
20580:
20533:
20514:(2): 223–231.
20498:
20461:(3): 563–572.
20445:
20404:
20369:
20354:
20327:
20321:hep-th/9312104
20306:
20284:
20268:Harris, Joseph
20255:
20243:
20235:
20208:
20196:
20184:
20171:
20169:
20166:
20165:
20164:
20159:
20154:
20149:
20132:, notably the
20122:
20117:
20112:
20101:
20095:
20092:Flag manifolds
20089:
20074:
20064:
20057:
20054:
19986:Schubert cells
19970:
19967:
19938:
19937:
19936:
19935:
19924:
19921:
19918:
19915:
19912:
19909:
19905:
19901:
19898:
19895:
19892:
19889:
19886:
19883:
19880:
19877:
19852:
19849:
19846:
19826:
19806:
19803:
19800:
19795:
19790:
19787:
19782:
19779:
19776:
19773:
19770:
19767:
19762:
19757:
19752:
19749:
19727:
19724:
19721:
19718:
19715:
19710:
19705:
19700:
19697:
19675:
19655:
19635:
19623:
19620:
19619:
19618:
19607:
19604:
19601:
19598:
19595:
19592:
19589:
19586:
19583:
19580:
19577:
19574:
19571:
19568:
19565:
19562:
19558:
19554:
19551:
19548:
19545:
19542:
19539:
19536:
19531:
19526:
19521:
19516:
19509:
19505:
19502:
19471:
19466:
19461:
19456:
19451:
19444:
19440:
19437:
19412:
19407:
19402:
19397:
19392:
19387:
19384:
19360:
19355:
19333:
19318:
19315:
19296:
19293:
19290:
19286:
19265:
19262:
19259:
19256:
19253:
19250:
19245:
19240:
19237:
19232:
19227:
19224:
19221:
19217:
19196:
19193:
19190:
19187:
19184:
19181:
19178:
19175:
19170:
19166:
19145:
19142:
19139:
19136:
19133:
19130:
19119:
19118:
19117:
19116:
19105:
19102:
19099:
19094:
19091:
19088:
19084:
19080:
19077:
19074:
19071:
19068:
19063:
19060:
19057:
19053:
19027:
19024:
19021:
19018:
19007:
19006:
19005:
19004:
18993:
18990:
18987:
18984:
18981:
18978:
18975:
18972:
18969:
18966:
18963:
18960:
18957:
18954:
18951:
18946:
18942:
18938:
18935:
18932:
18929:
18924:
18921:
18918:
18914:
18888:
18885:
18882:
18877:
18872:
18869:
18864:
18861:
18841:
18838:
18835:
18830:
18825:
18822:
18817:
18814:
18794:
18791:
18788:
18785:
18756:
18752:
18731:
18728:
18725:
18720:
18715:
18712:
18690:
18670:
18658:
18655:
18642:
18639:
18615:
18612:
18593:
18592:
18591:
18590:
18577:
18574:
18571:
18567:
18563:
18560:
18557:
18554:
18551:
18548:
18545:
18540:
18537:
18534:
18530:
18526:
18523:
18520:
18515:
18511:
18486:
18485:
18484:
18483:
18472:
18469:
18466:
18463:
18460:
18457:
18454:
18451:
18448:
18445:
18412:
18392:
18368:
18348:
18325:
18293:
18273:
18270:
18267:
18264:
18261:
18233:
18210:
18190:
18170:
18165:
18160:
18155:
18150:
18145:
18142:
18128:
18125:
18121:
18120:
18119:
18118:
18105:
18098:
18093:
18090:
18085:
18081:
18080:
18076:
18071:
18068:
18063:
18059:
18058:
18056:
18051:
18046:
18043:
18040:
18036:
18006:
17982:
17962:
17959:
17954:
17951:
17948:
17944:
17930:Solving these
17928:
17927:
17926:
17925:
17914:
17911:
17906:
17903:
17900:
17896:
17892:
17887:
17884:
17881:
17877:
17872:
17867:
17864:
17861:
17858:
17855:
17851:
17845:
17841:
17837:
17834:
17831:
17828:
17823:
17820:
17817:
17814:
17811:
17808:
17805:
17801:
17797:
17792:
17789:
17786:
17782:
17756:
17751:
17748:
17745:
17740:
17735:
17730:
17725:
17722:
17700:
17680:
17675:
17672:
17669:
17664:
17659:
17654:
17651:
17648:
17643:
17640:
17602:
17597:
17592:
17587:
17582:
17577:
17574:
17550:
17545:
17540:
17536:
17515:
17485:
17480:
17475:
17470:
17465:
17460:
17457:
17433:
17430:
17427:
17423:
17397:
17396:
17395:
17394:
17383:
17380:
17377:
17374:
17371:
17364:
17360:
17356:
17353:
17350:
17347:
17344:
17341:
17337:
17333:
17330:
17327:
17324:
17321:
17316:
17311:
17308:
17305:
17300:
17295:
17292:
17287:
17284:
17281:
17278:
17275:
17272:
17269:
17266:
17263:
17258:
17254:
17228:
17223:
17219:
17215:
17191:
17188:
17185:
17180:
17175:
17172:
17167:
17164:
17144:
17141:
17138:
17133:
17128:
17125:
17120:
17117:
17114:
17111:
17108:
17105:
17100:
17096:
17073:
17069:
17065:
17062:
17059:
17056:
17041:
17040:
17029:
17026:
17023:
17018:
17014:
17010:
17007:
17004:
16999:
16995:
16980:
16979:
16978:
16977:
16964:
16960:
16954:
16949:
16946:
16943:
16939:
16935:
16931:
16927:
16923:
16906:
16905:
16894:
16889:
16885:
16881:
16878:
16875:
16870:
16866:
16862:
16859:
16856:
16830:
16827:
16824:
16819:
16815:
16811:
16807:
16804:
16801:
16780:
16777:
16772:
16768:
16764:
16761:
16758:
16753:
16749:
16745:
16740:
16736:
16711:
16708:
16705:
16700:
16695:
16692:
16674:Schubert cells
16664:
16663:Schubert cells
16661:
16655:}-dimensional
16644:
16641:
16638:
16635:
16632:
16612:
16590:
16586:
16565:
16545:
16542:
16539:
16534:
16531:
16528:
16523:
16520:
16498:
16495:
16492:
16487:
16482:
16479:
16455:
16451:
16430:
16408:
16404:
16400:
16395:
16391:
16370:
16367:
16364:
16353:
16352:
16351:
16350:
16339:
16334:
16330:
16326:
16322:
16319:
16316:
16313:
16308:
16305:
16301:
16298:
16295:
16292:
16287:
16282:
16279:
16250:
16246:
16225:
16222:
16219:
16216:
16213:
16193:
16173:
16161:
16160:
16159:
16158:
16147:
16144:
16139:
16135:
16131:
16126:
16122:
16118:
16113:
16109:
16105:
16101:
16097:
16094:
16091:
16088:
16062:
16058:
16048:with quotient
16035:
16031:
16008:
16004:
16000:
15996:
15992:
15989:
15970:
15969:
15968:
15967:
15956:
15953:
15950:
15946:
15942:
15939:
15936:
15933:
15930:
15927:
15924:
15895:
15875:
15871:
15867:
15857:quotient space
15844:
15841:
15838:
15835:
15832:
15823:determines an
15812:
15809:
15806:
15786:
15774:
15771:
15758:
15755:
15750:
15746:
15742:
15738:
15717:
15714:
15711:
15708:
15703:
15698:
15695:
15690:
15687:
15676:
15675:
15674:
15673:
15662:
15659:
15654:
15650:
15644:
15640:
15636:
15631:
15627:
15621:
15617:
15613:
15608:
15604:
15598:
15594:
15568:
15563:
15559:
15555:
15550:
15546:
15542:
15537:
15533:
15529:
15524:
15520:
15516:
15511:
15507:
15503:
15498:
15494:
15490:
15470:
15467:
15462:
15458:
15454:
15450:
15446:
15443:
15440:
15437:
15432:
15427:
15424:
15419:
15416:
15396:
15393:
15390:
15370:
15367:
15364:
15361:
15358:
15355:
15352:
15329:
15324:
15320:
15299:
15296:
15293:
15290:
15264:
15260:
15250:with the term
15237:
15234:
15231:
15227:
15223:
15220:
15217:
15212:
15208:
15185:
15182:
15179:
15175:
15171:
15168:
15162:
15157:
15153:
15146:
15143:
15140:
15135:
15131:
15119:
15118:
15117:
15116:
15105:
15102:
15099:
15092:
15089:
15086:
15082:
15078:
15075:
15069:
15064:
15060:
15053:
15050:
15047:
15042:
15038:
15033:
15025:
15021:
15017:
15012:
15009:
15006:
15002:
14998:
14995:
14992:
14987:
14983:
14978:
14972:
14968:
14964:
14961:
14958:
14953:
14950:
14947:
14942:
14939:
14936:
14932:
14906:
14903:
14900:
14895:
14890:
14887:
14865:
14862:
14859:
14856:
14851:
14846:
14843:
14838:
14835:
14807:
14804:
14801:
14781:
14778:
14775:
14755:
14752:
14747:
14744:
14741:
14737:
14733:
14730:
14725:
14721:
14717:
14712:
14708:
14704:
14701:
14681:
14678:
14673:
14670:
14667:
14663:
14659:
14656:
14651:
14647:
14643:
14638:
14634:
14630:
14627:
14604:
14601:
14596:
14592:
14588:
14584:
14562:
14542:
14522:
14517:
14513:
14509:
14506:
14503:
14498:
14494:
14490:
14470:
14465:
14461:
14440:
14420:
14417:
14414:
14411:
14391:
14371:
14366:
14362:
14358:
14355:
14352:
14347:
14343:
14339:
14319:
14316:
14313:
14308:
14303:
14300:
14295:
14292:
14268:
14265:
14262:
14257:
14253:
14249:
14246:
14243:
14238:
14234:
14230:
14227:
14223:
14215:
14211:
14207:
14204:
14201:
14196:
14192:
14187:
14183:
14163:
14156:
14152:
14147:
14143:
14140:
14137:
14130:
14126:
14121:
14117:
14097:
14094:
14091:
14067:
14063:
14059:
14056:
14053:
14048:
14044:
14039:
14018:
13998:
13995:
13990:
13986:
13982:
13979:
13976:
13971:
13967:
13963:
13960:
13949:
13948:
13947:
13946:
13935:
13930:
13922:
13919:
13915:
13911:
13909:
13906:
13902:
13899:
13895:
13891:
13890:
13887:
13884:
13882:
13879:
13877:
13874:
13873:
13868:
13865:
13861:
13857:
13855:
13852:
13848:
13844:
13840:
13839:
13837:
13832:
13829:
13824:
13820:
13816:
13810:
13806:
13802:
13799:
13794:
13790:
13764:
13744:
13739:
13735:
13731:
13728:
13725:
13720:
13716:
13712:
13692:
13670:
13666:
13645:
13640:
13637:
13633:
13629:
13626:
13623:
13618:
13615:
13611:
13607:
13587:
13582:
13578:
13574:
13571:
13568:
13563:
13559:
13555:
13535:
13515:
13495:
13492:
13489:
13469:
13449:
13444:
13440:
13436:
13433:
13430:
13425:
13421:
13417:
13397:
13377:
13357:
13354:
13349:
13345:
13341:
13337:
13316:
13313:
13310:
13305:
13300:
13297:
13273:
13270:
13269:
13268:
13267:
13266:
13254:
13251:
13248:
13245:
13242:
13239:
13236:
13233:
13208:
13205:
13202:
13178:
13175:
13172:
13169:
13149:
13129:
13101:
13078:
13074:
13070:
13050:
13047:
13042:
13038:
13034:
13031:
13028:
13023:
13019:
13015:
13012:
13009:
13006:
13003:
13000:
12980:
12977:
12974:
12971:
12951:
12931:
12926:
12922:
12918:
12915:
12912:
12907:
12903:
12899:
12879:
12876:
12873:
12870:
12850:
12830:
12810:
12790:
12787:
12784:
12773:
12772:
12771:
12770:
12760:
12756:
12752:
12747:
12743:
12738:
12733:
12729:
12726:
12723:
12720:
12717:
12714:
12710:
12707:
12703:
12700:
12676:
12656:
12651:
12647:
12636:Exterior power
12623:
12603:
12600:
12597:
12594:
12591:
12587:
12584:
12564:Main article:
12561:
12558:
12557:
12556:
12555:
12554:
12543:
12539:
12535:
12532:
12529:
12526:
12523:
12520:
12517:
12514:
12509:
12504:
12501:
12498:
12494:
12491:
12487:
12484:
12481:
12478:
12475:
12472:
12469:
12465:
12461:
12458:
12455:
12452:
12449:
12446:
12442:
12425:
12424:
12423:
12422:
12412:
12407:
12402:
12399:
12396:
12392:
12389:
12383:
12379:
12375:
12372:
12369:
12365:
12340:
12337:
12334:
12331:
12326:
12321:
12317:
12296:
12293:
12290:
12286:
12265:
12262:
12259:
12239:
12236:
12233:
12230:
12225:
12220:
12217:
12214:
12210:
12207:
12186:
12164:
12142:
12122:
12119:
12116:
12094:
12091:
12086:
12082:
12078:
12073:
12068:
12064:
12043:
12023:
12020:
12017:
12014:
12009:
12004:
12001:
11998:
11994:
11991:
11970:
11948:
11942:
11930:-modules from
11917:
11913:
11897:
11896:
11895:
11894:
11883:
11880:
11875:
11871:
11867:
11862:
11857:
11853:
11849:
11846:
11841:
11835:
11829:
11825:
11808:
11807:
11806:
11805:
11794:
11791:
11786:
11781:
11778:
11775:
11771:
11768:
11764:
11761:
11741:
11740:
11739:
11738:
11727:
11724:
11719:
11714:
11711:
11708:
11704:
11701:
11695:
11691:
11687:
11682:
11677:
11673:
11669:
11665:
11660:
11655:
11650:
11647:
11644:
11640:
11637:
11630:
11624:
11619:
11615:
11612:
11607:
11602:
11598:
11568:
11563:
11558:
11555:
11552:
11548:
11545:
11524:
11502:
11497:
11492:
11489:
11486:
11482:
11479:
11472:
11447:
11425:
11421:
11415:
11410:
11405:
11402:
11399:
11395:
11392:
11385:
11379:
11376:
11372:
11367:
11364:
11341:
11327:
11324:
11311:
11308:
11305:
11302:
11279:
11276:
11273:
11270:
11267:
11260:
11256:
11251:
11245:
11240:
11237:
11234:
11230:
11227:
11204:
11200:
11194:
11189:
11186:
11183:
11179:
11176:
11155:
11152:
11149:
11146:
11143:
11140:
11132:
11129:
11126:
11123:
11103:
11083:
11072:
11071:
11070:
11069:
11058:
11052:
11049:
11042:
11036:
11033:
11030:
11026:
11023:
11019:
11015:
11012:
11006:
11002:
10998:
10993:
10988:
10985:
10982:
10978:
10975:
10948:
10945:
10924:
10904:
10901:
10898:
10895:
10892:
10888:
10885:
10864:
10840:
10818:
10796:
10776:
10754:
10728:
10723:
10718:
10715:
10712:
10708:
10705:
10684:
10657:
10652:
10646:
10640:
10637:
10634:
10630:
10627:
10606:
10586:
10575:
10574:
10573:
10572:
10559:
10555:
10547:
10543:
10538:
10532:
10527:
10522:
10516:
10484:
10464:
10444:
10424:
10398:
10384:
10381:
10364:
10361:
10338:
10337:
10326:
10322:
10318:
10315:
10312:
10309:
10306:
10303:
10300:
10297:
10294:
10291:
10288:
10284:
10279:
10275:
10272:
10269:
10266:
10263:
10260:
10255:
10250:
10245:
10242:
10239:
10235:
10232:
10206:
10201:
10196:
10191:
10186:
10181:
10176:
10172:
10149:
10144:
10139:
10136:
10125:
10124:
10113:
10109:
10105:
10098:
10093:
10089:
10084:
10080:
10076:
10070:
10065:
10061:
10057:
10054:
10051:
10048:
10041:
10037:
10031:
10026:
10023:
10018:
10014:
10010:
10007:
10003:
9998:
9994:
9991:
9988:
9985:
9982:
9979:
9976:
9973:
9970:
9967:
9964:
9961:
9957:
9954:
9930:
9927:
9924:
9921:
9918:
9915:
9892:
9875:
9874:
9863:
9859:
9855:
9852:
9849:
9846:
9843:
9840:
9837:
9834:
9831:
9828:
9825:
9821:
9816:
9812:
9809:
9806:
9803:
9800:
9797:
9792:
9787:
9782:
9779:
9776:
9772:
9769:
9745:
9723:
9718:
9713:
9708:
9703:
9698:
9693:
9689:
9666:
9661:
9656:
9653:
9642:
9641:
9628:
9624:
9617:
9613:
9607:
9602:
9599:
9594:
9590:
9586:
9583:
9580:
9577:
9572:
9567:
9562:
9559:
9556:
9553:
9550:
9546:
9541:
9537:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9500:
9497:
9473:
9451:
9447:
9424:
9419:
9415:
9403:
9402:
9390:
9383:
9378:
9374:
9368:
9363:
9360:
9355:
9350:
9346:
9342:
9339:
9336:
9333:
9326:
9322:
9316:
9311:
9308:
9303:
9299:
9295:
9292:
9269:
9266:
9261:
9257:
9236:
9216:
9213:
9210:
9207:
9204:
9200:
9197:
9176:
9156:
9153:
9150:
9147:
9144:
9141:
9118:
9098:
9077:
9055:
9033:
9009:
9006:
9003:
8999:
8996:
8971:
8932:
8929:
8926:
8922:
8919:
8898:
8867:
8864:
8861:
8857:
8854:
8832:
8810:
8786:
8771:
8770:
8759:
8755:
8751:
8748:
8745:
8741:
8738:
8734:
8731:
8728:
8725:
8722:
8719:
8715:
8712:
8688:
8685:
8682:
8679:
8676:
8672:
8669:
8645:
8642:
8639:
8635:
8632:
8620:
8619:
8608:
8605:
8602:
8598:
8595:
8591:
8588:
8583:
8579:
8575:
8572:
8567:
8563:
8559:
8556:
8551:
8546:
8543:
8540:
8536:
8533:
8529:
8526:
8523:
8520:
8517:
8512:
8508:
8504:
8500:
8497:
8494:
8491:
8487:
8484:
8459:
8455:
8430:
8427:
8424:
8421:
8418:
8398:
8378:
8375:
8372:
8368:
8365:
8361:
8358:
8347:
8346:
8335:
8330:
8326:
8322:
8319:
8316:
8313:
8290:
8287:
8284:
8281:
8278:
8274:
8271:
8267:
8264:
8255:, any element
8244:
8224:
8221:
8216:
8212:
8191:
8171:
8145:
8142:
8139:
8135:
8132:
8115:
8112:
8111:
8110:
8109:
8108:
8097:
8092:
8087:
8083:
8077:
8073:
8067:
8062:
8059:
8056:
8052:
8048:
8043:
8039:
8013:
8008:
8004:
8000:
7997:
7994:
7989:
7985:
7981:
7959:
7954:
7949:
7946:
7926:
7904:
7900:
7879:
7874:
7870:
7866:
7863:
7860:
7855:
7851:
7847:
7827:
7823:
7820:
7817:
7813:
7798:
7797:
7783:
7780:
7777:
7774:
7771:
7767:
7764:
7741:
7731:
7719:
7716:
7711:
7707:
7682:
7672:
7660:
7657:
7652:
7648:
7623:
7600:
7596:
7592:
7589:
7586:
7583:
7580:
7577:
7557:
7554:
7551:
7531:
7527:
7523:
7520:
7517:
7514:
7511:
7508:
7504:
7500:
7497:
7494:
7491:
7488:
7485:
7465:
7460:
7455:
7450:
7447:
7444:
7440:
7437:
7413:
7391:
7386:
7381:
7378:
7358:
7353:
7349:
7345:
7342:
7339:
7334:
7330:
7326:
7315:
7314:
7313:
7312:
7301:
7296:
7291:
7287:
7281:
7277:
7271:
7266:
7263:
7260:
7256:
7252:
7247:
7243:
7217:
7212:
7207:
7202:
7199:
7196:
7192:
7189:
7185:
7182:
7160:
7155:
7133:
7113:
7109:
7105:
7102:
7099:
7096:
7093:
7090:
7087:
7084:
7062:
7057:
7035:
7015:
7010:
7005:
7000:
6997:
6994:
6990:
6987:
6966:
6962:
6958:
6955:
6952:
6949:
6946:
6943:
6932:
6931:
6919:
6916:
6913:
6910:
6907:
6904:
6901:
6881:
6871:
6859:
6856:
6851:
6847:
6822:
6812:
6800:
6797:
6792:
6788:
6763:
6740:
6736:
6732:
6729:
6726:
6723:
6720:
6717:
6697:
6693:
6689:
6686:
6683:
6680:
6677:
6674:
6670:
6666:
6663:
6660:
6657:
6654:
6651:
6631:
6628:
6625:
6605:
6601:
6597:
6594:
6591:
6588:
6576:
6551:Grassmannians
6549:
6533:
6501:
6481:
6478:
6474:
6471:
6467:
6464:
6453:
6452:
6451:
6450:
6439:
6436:
6430:
6427:
6422:
6418:
6413:
6409:
6405:
6402:
6399:
6395:
6392:
6388:
6385:
6382:
6379:
6350:
6347:
6344:
6341:
6338:
6334:
6331:
6303:
6300:
6297:
6294:
6289:
6285:
6281:
6278:
6274:
6271:
6267:
6264:
6261:
6258:
6254:
6251:
6248:
6244:
6241:
6238:
6213:
6210:
6207:
6202:
6178:
6175:
6172:
6167:
6145:
6142:
6139:
6119:
6114:
6109:
6104:
6101:
6098:
6094:
6091:
6070:
6065:
6060:
6055:
6052:
6049:
6045:
6042:
6019:
6014:
6009:
6006:
5984:
5979:
5974:
5971:
5960:
5959:
5958:
5957:
5946:
5942:
5938:
5935:
5932:
5929:
5926:
5922:
5919:
5914:
5909:
5905:
5901:
5896:
5892:
5888:
5885:
5882:
5879:
5876:
5873:
5869:
5866:
5863:
5859:
5856:
5852:
5848:
5845:
5842:
5839:
5836:
5833:
5829:
5826:
5800:
5780:
5760:
5757:
5754:
5751:
5748:
5744:
5741:
5716:
5713:
5710:
5706:
5703:
5699:
5694:
5690:
5669:
5649:
5638:
5637:
5636:
5635:
5622:
5617:
5612:
5608:
5604:
5601:
5596: if
5592:
5589:
5588:
5585:
5582:
5579:
5574: if
5570:
5567:
5566:
5564:
5559:
5556:
5553:
5550:
5545:
5541:
5513:
5509:
5488:
5477:
5476:
5475:
5474:
5461:
5457:
5453:
5450:
5447:
5444:
5419:
5399:
5396:
5393:
5369:
5366:
5363:
5360:
5355:
5351:
5330:
5310:
5290:
5270:
5250:
5247:
5244:
5241:
5238:
5215:
5212:
5199:
5196:
5193:
5188:
5183:
5180:
5158:
5151:
5147:
5143:
5140:
5137:
5132:
5128:
5123:
5119:
5112:
5108:
5104:
5101:
5098:
5093:
5089:
5084:
5080:
5056:
5052:
5048:
5045:
5042:
5037:
5033:
5028:
5003:
4999:
4995:
4992:
4989:
4984:
4980:
4972:
4969:
4943:
4939:
4929:th row is the
4918:
4898:
4895:
4892:
4868:
4864:
4860:
4857:
4854:
4849:
4845:
4837:
4833:
4829:
4826:
4823:
4818:
4814:
4806:
4803:
4788:
4787:
4786:
4785:
4774:
4769:
4766:
4762:
4754:
4750:
4746:
4743:
4740:
4735:
4731:
4723:
4719:
4715:
4712:
4709:
4704:
4700:
4692:
4689:
4682:
4675:
4671:
4667:
4664:
4661:
4656:
4652:
4644:
4641:
4634:
4627:
4623:
4619:
4616:
4613:
4608:
4604:
4596:
4593:
4560:
4556:
4552:
4549:
4546:
4541:
4537:
4532:
4507:
4503:
4499:
4496:
4493:
4488:
4484:
4479:
4467:
4466:
4465:
4464:
4453:
4446:
4442:
4438:
4435:
4432:
4427:
4423:
4418:
4410:
4406:
4402:
4399:
4396:
4391:
4387:
4379:
4376:
4369:
4362:
4358:
4354:
4351:
4348:
4343:
4339:
4334:
4326:
4322:
4318:
4315:
4312:
4307:
4303:
4295:
4292:
4259:
4255:
4251:
4248:
4245:
4240:
4236:
4231:
4206:
4202:
4198:
4195:
4192:
4187:
4183:
4178:
4153:
4149:
4145:
4142:
4139:
4134:
4130:
4125:
4121:
4114:
4110:
4106:
4103:
4100:
4095:
4091:
4086:
4061:
4057:
4053:
4050:
4047:
4042:
4038:
4033:
4012:
4007:
4003:
3999:
3996:
3993:
3988:
3984:
3980:
3960:
3957:
3954:
3943:
3942:
3941:
3940:
3927:
3924:
3920:
3912:
3908:
3904:
3901:
3898:
3893:
3889:
3884:
3880:
3877:
3874:
3867:
3863:
3859:
3856:
3853:
3848:
3844:
3836:
3833:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3766:
3742:
3738:
3734:
3731:
3728:
3723:
3719:
3714:
3702:diffeomorphism
3685:
3681:
3677:
3674:
3671:
3666:
3662:
3657:
3632:
3628:
3624:
3621:
3618:
3613:
3609:
3604:
3575:
3571:
3567:
3564:
3561:
3556:
3552:
3547:
3526:
3523:
3520:
3515:
3510:
3507:
3502:
3499:
3479:
3476:
3473:
3453:
3433:
3430:
3427:
3407:
3402:
3398:
3394:
3391:
3388:
3383:
3379:
3375:
3353:
3350:
3343:
3339:
3335:
3332:
3329:
3324:
3320:
3315:
3311:
3287:
3283:
3279:
3276:
3273:
3268:
3264:
3259:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3196:
3192:
3188:
3185:
3182:
3177:
3173:
3168:
3147:
3125:
3121:
3100:
3076:
3072:
3068:
3065:
3062:
3057:
3053:
3048:
3027:
3024:
3021:
3001:
2981:
2978:
2975:
2970:
2965:
2962:
2957:
2954:
2930:
2926:
2922:
2919:
2916:
2911:
2907:
2902:
2881:
2878:
2873:
2869:
2865:
2862:
2859:
2854:
2850:
2846:
2843:
2820:
2792:
2772:
2767:
2763:
2759:
2756:
2753:
2748:
2744:
2740:
2716:
2712:
2708:
2705:
2702:
2697:
2693:
2688:
2667:
2664:
2661:
2652:such that the
2641:
2638:
2633:
2629:
2625:
2622:
2619:
2614:
2610:
2606:
2603:
2583:
2563:
2543:
2523:
2503:
2498:
2495:
2491:
2487:
2467:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2418:
2417:
2416:
2415:
2402:
2394:
2391:
2388:
2385:
2382:
2378:
2374:
2372:
2369:
2367:
2364:
2360:
2357:
2354:
2351:
2348:
2344:
2340:
2339:
2336:
2333:
2329:
2326:
2325:
2320:
2317:
2314:
2310:
2306:
2304:
2301:
2299:
2296:
2292:
2289:
2286:
2282:
2278:
2277:
2274:
2271:
2267:
2264:
2261:
2258:
2255:
2252:
2250:
2247:
2244:
2243:
2241:
2214:
2194:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2126:
2102:
2082:
2079:
2076:
2053:
2050:
2047:
2027:
2007:
1987:
1984:
1981:
1961:
1941:
1938:
1935:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1873:
1872:
1871:
1870:
1859:
1856:
1853:
1847:
1844:
1816:
1813:
1810:
1805:
1800:
1797:
1792:
1789:
1766:
1763:
1740:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1680:
1676:
1655:
1635:
1608:
1605:
1602:
1582:
1579:
1576:
1571:
1566:
1563:
1558:
1555:
1531:
1526:
1522:
1518:
1515:
1512:
1507:
1503:
1499:
1476:
1456:
1453:
1450:
1445:
1440:
1437:
1415:
1412:
1409:
1389:
1369:
1364:
1360:
1356:
1353:
1350:
1345:
1341:
1337:
1315:
1311:
1290:
1270:
1246:
1243:
1240:
1235:
1230:
1227:
1213:
1210:
1140:
1137:
1134:
1129:
1090:
1089:Low dimensions
1087:
1058:
1038:
1027:vector bundles
1007:
987:
967:
964:
961:
941:
936:
931:
926:
921:
916:
913:
887:
861:
856:
832:
827:
805:
781:
778:
775:
753:
748:
726:
706:
663:
660:
647:
627:
607:
587:
584:
581:
578:
575:
571:
568:
547:
544:
541:
536:
531:
528:
506:
503:
500:
497:
494:
490:
487:
466:
463:
460:
455:
450:
447:
410:
405:
400:
395:
390:
385:
382:
369:Julius Plücker
349:
346:
343:
340:
337:
334:
299:
279:
259:
256:
253:
249:
225:
205:
202:
199:
194:
189:
186:
164:
141:
118:
91:
63:
60:
57:
52:
47:
44:
15:
9:
6:
4:
3:
2:
21358:
21347:
21344:
21342:
21339:
21337:
21334:
21332:
21329:
21328:
21326:
21315:
21309:
21305:
21301:
21297:
21293:
21292:
21287:
21283:
21279:
21277:0-521-65595-1
21273:
21269:
21268:
21263:
21259:
21255:
21251:
21247:
21241:
21237:
21233:
21229:
21225:
21221:
21218:
21212:
21208:
21203:
21199:
21197:0-387-97716-3
21193:
21189:
21188:
21182:
21177:
21175:0-691-08122-0
21171:
21167:
21163:
21159:
21155:
21147:
21146:
21141:
21137:
21133:
21129:
21125:
21121:
21117:
21115:0-471-05059-8
21111:
21107:
21103:
21099:
21095:
21091:
21090:
21071:
21067:
21063:
21059:
21055:
21051:
21047:
21043:
21039:
21035:
21034:
21026:
21022:
21015:
21008:
21004:
21000:
20996:
20992:
20986:
20978:
20974:
20970:
20966:
20962:
20958:
20953:
20948:
20944:
20940:
20933:
20918:
20914:
20907:
20899:
20895:
20891:
20887:
20883:
20879:
20874:
20869:
20865:
20861:
20854:
20846:
20842:
20838:
20834:
20829:
20824:
20820:
20816:
20809:
20801:
20797:
20793:
20791:9781108610902
20787:
20783:
20779:
20775:
20771:
20765:
20757:
20753:
20749:
20745:
20741:
20734:
20726:
20722:
20718:
20716:9781108610902
20712:
20708:
20704:
20700:
20696:
20690:
20688:
20679:
20675:
20670:
20665:
20661:
20657:
20653:
20646:
20644:
20635:
20631:
20627:
20623:
20619:
20615:
20611:
20607:
20600:
20598:
20590:
20584:
20575:
20570:
20565:
20560:
20556:
20552:
20548:
20544:
20543:Varchenko, A.
20537:
20529:
20525:
20521:
20517:
20513:
20509:
20502:
20494:
20490:
20486:
20482:
20477:
20472:
20468:
20464:
20460:
20456:
20449:
20440:
20435:
20431:
20427:
20423:
20419:
20415:
20408:
20400:
20396:
20392:
20388:
20384:
20380:
20373:
20365:
20361:
20357:
20351:
20347:
20343:
20342:
20334:
20332:
20322:
20317:
20310:
20303:
20299:
20295:
20291:
20287:
20285:0-471-05059-8
20281:
20277:
20273:
20269:
20265:
20259:
20252:
20247:
20241:, Chapter I.9
20238:
20232:
20228:
20224:
20223:
20218:
20212:
20205:
20200:
20193:
20188:
20181:
20176:
20172:
20163:
20160:
20158:
20155:
20153:
20150:
20147:
20143:
20139:
20137:
20131:
20127:
20123:
20121:
20118:
20116:
20113:
20110:
20106:
20102:
20099:
20096:
20093:
20090:
20087:
20083:
20079:
20075:
20073:
20069:
20065:
20063:
20060:
20059:
20053:
20051:
20047:
20042:
20040:
20039:amplituhedron
20036:
20032:
20028:
20023:
20021:
20016:
20014:
20010:
20006:
20002:
19997:
19995:
19991:
19987:
19983:
19979:
19974:
19966:
19964:
19960:
19956:
19952:
19951:
19946:
19942:
19922:
19919:
19916:
19913:
19910:
19907:
19899:
19896:
19893:
19887:
19884:
19881:
19875:
19868:
19867:
19866:
19865:
19864:
19850:
19847:
19844:
19824:
19801:
19793:
19780:
19774:
19771:
19768:
19760:
19755:
19722:
19719:
19716:
19708:
19703:
19673:
19653:
19633:
19605:
19596:
19593:
19590:
19584:
19581:
19578:
19572:
19566:
19563:
19556:
19549:
19543:
19540:
19537:
19529:
19514:
19507:
19488:
19487:
19486:
19483:
19464:
19449:
19442:
19405:
19390:
19358:
19331:
19324:
19314:
19312:
19311:Radon measure
19294:
19291:
19288:
19284:
19276:. Moreover,
19263:
19260:
19251:
19243:
19225:
19222:
19219:
19215:
19194:
19191:
19182:
19176:
19168:
19164:
19140:
19134:
19131:
19128:
19100:
19092:
19089:
19086:
19082:
19078:
19072:
19069:
19061:
19058:
19055:
19051:
19043:
19042:
19041:
19040:
19039:
19022:
19016:
18991:
18985:
18982:
18979:
18976:
18973:
18967:
18961:
18955:
18952:
18944:
18940:
18936:
18930:
18922:
18919:
18916:
18912:
18904:
18903:
18902:
18901:
18900:
18883:
18875:
18862:
18859:
18836:
18828:
18815:
18812:
18789:
18783:
18776:
18772:
18754:
18750:
18726:
18718:
18688:
18668:
18654:
18640:
18637:
18629:
18613:
18610:
18602:
18598:
18575:
18572:
18569:
18561:
18558:
18552:
18546:
18538:
18535:
18532:
18528:
18521:
18513:
18509:
18501:
18500:
18499:
18498:
18497:
18495:
18494:Edward Witten
18491:
18470:
18467:
18461:
18455:
18449:
18443:
18436:
18435:
18434:
18433:
18432:
18430:
18426:
18425:Functoriality
18423:is trivial.
18410:
18390:
18382:
18366:
18346:
18337:
18323:
18315:
18314:Chern classes
18311:
18307:
18291:
18268:
18265:
18262:
18251:
18247:
18231:
18224:
18223:vector bundle
18208:
18188:
18163:
18148:
18124:
18103:
18096:
18091:
18088:
18083:
18074:
18069:
18066:
18061:
18054:
18049:
18044:
18041:
18038:
18034:
18026:
18025:
18024:
18023:
18022:
18020:
18004:
17996:
17980:
17960:
17957:
17952:
17949:
17946:
17942:
17933:
17912:
17909:
17904:
17901:
17898:
17894:
17890:
17885:
17882:
17879:
17875:
17870:
17865:
17862:
17859:
17856:
17853:
17849:
17843:
17835:
17832:
17826:
17821:
17818:
17815:
17812:
17809:
17806:
17803:
17799:
17795:
17790:
17787:
17784:
17780:
17772:
17771:
17770:
17769:
17768:
17749:
17746:
17743:
17728:
17698:
17673:
17670:
17667:
17652:
17649:
17646:
17627:
17623:that contain
17621:
17595:
17580:
17548:
17538:
17513:
17504:
17478:
17463:
17431:
17428:
17425:
17421:
17413:
17408:
17406:
17402:
17381:
17375:
17372:
17362:
17358:
17354:
17351:
17348:
17345:
17342:
17339:
17335:
17331:
17328:
17322:
17319:
17306:
17298:
17285:
17282:
17276:
17270:
17267:
17264:
17256:
17252:
17244:
17243:
17242:
17241:
17240:
17221:
17217:
17205:
17204:intersections
17186:
17178:
17165:
17162:
17139:
17131:
17118:
17112:
17109:
17106:
17098:
17094:
17071:
17063:
17060:
17057:
17046:
17045:Young diagram
17027:
17024:
17021:
17016:
17012:
17008:
17005:
17002:
16997:
16993:
16985:
16984:
16983:
16962:
16958:
16952:
16947:
16944:
16941:
16937:
16933:
16925:
16913:
16912:
16911:
16910:
16909:
16887:
16883:
16879:
16876:
16873:
16868:
16864:
16857:
16854:
16847:
16846:
16845:
16844:
16828:
16825:
16817:
16813:
16791:of dimension
16778:
16775:
16770:
16766:
16762:
16759:
16756:
16751:
16747:
16743:
16738:
16734:
16726:of subspaces
16725:
16724:complete flag
16706:
16698:
16680:
16676:
16675:
16670:
16660:
16658:
16639:
16636:
16633:
16610:
16588:
16584:
16563:
16540:
16532:
16529:
16526:
16493:
16485:
16453:
16449:
16428:
16406:
16402:
16398:
16393:
16389:
16368:
16365:
16362:
16332:
16328:
16324:
16320:
16317:
16314:
16293:
16285:
16268:
16267:
16266:
16265:
16264:
16248:
16244:
16220:
16217:
16214:
16191:
16171:
16145:
16137:
16133:
16124:
16120:
16111:
16103:
16099:
16095:
16086:
16079:
16078:
16077:
16076:
16075:
16060:
16056:
16033:
16029:
16006:
15998:
15994:
15990:
15979:
15975:
15954:
15948:
15944:
15940:
15934:
15928:
15922:
15915:
15914:
15913:
15912:
15911:
15909:
15893:
15873:
15869:
15865:
15858:
15855:-dimensional
15839:
15836:
15833:
15810:
15807:
15804:
15784:
15770:
15753:
15748:
15709:
15701:
15685:
15660:
15657:
15652:
15648:
15642:
15638:
15634:
15629:
15625:
15619:
15615:
15611:
15606:
15602:
15596:
15592:
15584:
15583:
15582:
15581:
15580:
15561:
15557:
15553:
15548:
15544:
15540:
15535:
15531:
15527:
15522:
15518:
15514:
15509:
15505:
15501:
15496:
15492:
15465:
15460:
15444:
15438:
15430:
15414:
15394:
15391:
15388:
15368:
15365:
15359:
15353:
15350:
15341:
15327:
15322:
15294:
15288:
15280:
15262:
15258:
15235:
15232:
15229:
15225:
15221:
15218:
15215:
15210:
15206:
15183:
15180:
15177:
15173:
15169:
15166:
15160:
15155:
15151:
15144:
15141:
15138:
15133:
15129:
15103:
15100:
15097:
15090:
15087:
15084:
15080:
15076:
15073:
15067:
15062:
15058:
15051:
15048:
15045:
15040:
15036:
15031:
15023:
15019:
15015:
15010:
15007:
15004:
15000:
14996:
14993:
14990:
14985:
14981:
14976:
14970:
14962:
14959:
14951:
14948:
14945:
14940:
14937:
14934:
14930:
14922:
14921:
14920:
14919:
14918:
14901:
14893:
14857:
14849:
14833:
14825:
14821:
14805:
14802:
14799:
14779:
14776:
14773:
14753:
14750:
14745:
14742:
14739:
14735:
14731:
14728:
14723:
14719:
14715:
14710:
14706:
14702:
14699:
14679:
14676:
14671:
14668:
14665:
14661:
14657:
14654:
14649:
14645:
14641:
14636:
14632:
14628:
14625:
14616:
14599:
14594:
14560:
14540:
14515:
14511:
14507:
14504:
14501:
14496:
14492:
14468:
14463:
14438:
14415:
14409:
14389:
14364:
14360:
14356:
14353:
14350:
14345:
14341:
14314:
14306:
14293:
14290:
14282:
14263:
14260:
14255:
14251:
14247:
14244:
14241:
14236:
14232:
14228:
14225:
14213:
14209:
14205:
14202:
14199:
14194:
14190:
14185:
14154:
14150:
14145:
14141:
14138:
14135:
14128:
14124:
14119:
14095:
14092:
14089:
14065:
14061:
14057:
14054:
14051:
14046:
14042:
14037:
14016:
13996:
13993:
13988:
13984:
13980:
13977:
13974:
13969:
13965:
13961:
13958:
13933:
13928:
13920:
13917:
13913:
13907:
13900:
13897:
13893:
13885:
13880:
13875:
13866:
13863:
13859:
13853:
13846:
13842:
13835:
13830:
13822:
13818:
13814:
13808:
13804:
13797:
13792:
13788:
13780:
13779:
13778:
13777:
13776:
13762:
13737:
13733:
13729:
13726:
13723:
13718:
13714:
13690:
13668:
13664:
13638:
13635:
13631:
13627:
13624:
13621:
13616:
13613:
13609:
13580:
13576:
13572:
13569:
13566:
13561:
13557:
13533:
13513:
13493:
13490:
13487:
13467:
13442:
13438:
13434:
13431:
13428:
13423:
13419:
13395:
13375:
13352:
13347:
13311:
13303:
13285:
13284:
13279:
13252:
13249:
13243:
13237:
13234:
13231:
13224:
13223:
13222:
13221:
13220:
13206:
13203:
13200:
13192:
13173:
13167:
13147:
13127:
13119:
13115:
13099:
13090:
13072:
13048:
13040:
13036:
13032:
13029:
13026:
13021:
13017:
13010:
13004:
12998:
12975:
12969:
12949:
12924:
12920:
12916:
12913:
12910:
12905:
12901:
12874:
12868:
12861:. To define
12848:
12828:
12808:
12788:
12785:
12782:
12775:Suppose that
12758:
12754:
12750:
12745:
12736:
12721:
12718:
12715:
12701:
12698:
12692:
12691:
12690:
12689:
12688:
12674:
12654:
12649:
12637:
12621:
12598:
12595:
12592:
12573:
12567:
12541:
12537:
12533:
12530:
12527:
12524:
12518:
12502:
12499:
12485:
12479:
12470:
12459:
12453:
12450:
12447:
12440:
12432:
12431:
12430:
12429:
12428:
12400:
12397:
12381:
12377:
12370:
12356:
12355:
12354:
12353:
12352:
12335:
12291:
12263:
12260:
12257:
12234:
12218:
12215:
12184:
12140:
12120:
12117:
12114:
12105:
12092:
12089:
12084:
12080:
12041:
12018:
12002:
11999:
11968:
11946:
11915:
11911:
11902:
11881:
11878:
11873:
11869:
11839:
11815:
11814:
11813:
11812:
11811:
11792:
11779:
11776:
11759:
11752:
11751:
11750:
11749:
11748:
11725:
11712:
11709:
11693:
11689:
11667:
11663:
11648:
11645:
11622:
11588:
11587:
11586:
11585:
11584:
11582:
11556:
11553:
11522:
11490:
11487:
11423:
11419:
11403:
11400:
11377:
11374:
11370:
11323:
11306:
11300:
11293:
11292:residue field
11271:
11265:
11258:
11254:
11249:
11238:
11235:
11202:
11187:
11184:
11153:
11144:
11138:
11130:
11124:
11101:
11081:
11050:
11047:
11034:
11031:
11017:
11013:
11010:
11004:
11000:
10986:
10983:
10965:
10964:
10963:
10962:
10961:
10946:
10943:
10922:
10899:
10896:
10893:
10862:
10854:
10838:
10794:
10774:
10742:
10716:
10713:
10682:
10674:
10669:
10650:
10638:
10635:
10604:
10584:
10557:
10553:
10545:
10541:
10536:
10525:
10520:
10504:
10503:
10502:
10501:
10500:
10498:
10482:
10462:
10442:
10422:
10414:
10380:
10378:
10374:
10370:
10360:
10356:
10352:
10348:
10343:
10324:
10320:
10313:
10310:
10307:
10301:
10298:
10292:
10286:
10282:
10277:
10270:
10264:
10261:
10253:
10243:
10240:
10222:
10221:
10220:
10204:
10194:
10189:
10179:
10174:
10170:
10147:
10137:
10134:
10111:
10107:
10096:
10091:
10087:
10082:
10078:
10068:
10063:
10059:
10052:
10049:
10039:
10035:
10024:
10021:
10016:
10012:
10005:
10001:
9996:
9989:
9986:
9983:
9977:
9974:
9968:
9965:
9962:
9944:
9943:
9942:
9925:
9922:
9919:
9913:
9906:
9905:unitary group
9890:
9881:
9861:
9857:
9850:
9847:
9844:
9838:
9835:
9829:
9823:
9819:
9814:
9807:
9801:
9798:
9790:
9780:
9777:
9759:
9758:
9757:
9743:
9721:
9711:
9706:
9696:
9691:
9687:
9664:
9654:
9651:
9626:
9615:
9611:
9600:
9597:
9592:
9588:
9581:
9578:
9570:
9560:
9557:
9554:
9548:
9544:
9539:
9532:
9529:
9526:
9520:
9517:
9511:
9508:
9505:
9487:
9486:
9485:
9471:
9449:
9445:
9422:
9417:
9413:
9381:
9376:
9372:
9361:
9358:
9353:
9348:
9344:
9337:
9334:
9324:
9320:
9309:
9306:
9301:
9297:
9290:
9283:
9282:
9281:
9267:
9264:
9259:
9255:
9234:
9211:
9208:
9205:
9174:
9151:
9148:
9145:
9139:
9132:
9116:
9096:
9021:
9004:
8985:
8969:
8961:
8957:
8953:
8950:
8946:
8927:
8896:
8889:
8885:
8881:
8862:
8798:
8784:
8776:
8757:
8753:
8746:
8732:
8726:
8723:
8720:
8702:
8701:
8700:
8683:
8680:
8677:
8657:
8640:
8603:
8589:
8581:
8577:
8573:
8565:
8561:
8554:
8541:
8527:
8524:
8518:
8510:
8506:
8485:
8482:
8475:
8474:
8473:
8457:
8453:
8444:
8425:
8422:
8419:
8396:
8373:
8359:
8356:
8328:
8324:
8317:
8314:
8311:
8304:
8303:
8302:
8285:
8282:
8279:
8265:
8262:
8242:
8235:of dimension
8222:
8219:
8214:
8210:
8189:
8169:
8161:
8158:
8140:
8121:
8095:
8090:
8085:
8081:
8075:
8071:
8065:
8060:
8057:
8054:
8050:
8046:
8041:
8037:
8029:
8028:
8027:
8026:
8025:
8006:
8002:
7998:
7995:
7992:
7987:
7983:
7957:
7947:
7944:
7924:
7902:
7898:
7872:
7868:
7864:
7861:
7858:
7853:
7849:
7821:
7818:
7815:
7803:
7778:
7739:
7732:
7717:
7714:
7709:
7705:
7697:(Hermitian):
7696:
7680:
7673:
7658:
7655:
7650:
7646:
7637:
7621:
7614:
7613:
7612:
7590:
7587:
7581:
7578:
7575:
7555:
7552:
7549:
7521:
7518:
7512:
7509:
7498:
7495:
7492:
7489:
7483:
7458:
7448:
7445:
7425:
7411:
7389:
7379:
7376:
7351:
7347:
7343:
7340:
7337:
7332:
7328:
7299:
7294:
7289:
7285:
7279:
7275:
7269:
7264:
7261:
7258:
7254:
7250:
7245:
7241:
7233:
7232:
7231:
7230:
7229:
7210:
7200:
7197:
7183:
7180:
7158:
7131:
7103:
7100:
7097:
7094:
7088:
7085:
7082:
7060:
7033:
7008:
6998:
6995:
6956:
6953:
6950:
6947:
6941:
6917:
6914:
6908:
6902:
6899:
6879:
6872:
6857:
6854:
6849:
6845:
6836:
6820:
6813:
6798:
6795:
6790:
6786:
6777:
6761:
6754:
6753:
6752:
6730:
6727:
6721:
6718:
6715:
6687:
6684:
6678:
6675:
6664:
6661:
6658:
6655:
6649:
6629:
6626:
6623:
6595:
6592:
6586:
6574:
6570:
6566:
6562:
6558:
6554:
6548:
6545:
6531:
6523:
6522:operator norm
6499:
6479:
6476:
6472:
6469:
6465:
6462:
6455:for any pair
6437:
6428:
6425:
6420:
6416:
6411:
6407:
6400:
6393:
6390:
6386:
6383:
6377:
6370:
6369:
6368:
6367:
6366:
6364:
6345:
6342:
6339:
6320:
6317:
6298:
6295:
6287:
6283:
6279:
6265:
6259:
6242:
6239:
6227:
6211:
6208:
6205:
6176:
6173:
6170:
6143:
6140:
6137:
6112:
6102:
6099:
6063:
6053:
6050:
6017:
6007:
6004:
5982:
5972:
5969:
5944:
5940:
5936:
5933:
5927:
5912:
5907:
5903:
5899:
5894:
5890:
5886:
5883:
5880:
5874:
5857:
5854:
5850:
5846:
5840:
5837:
5834:
5816:
5815:
5814:
5813:
5812:
5798:
5778:
5755:
5752:
5749:
5730:
5711:
5697:
5692:
5688:
5667:
5647:
5615:
5610:
5606:
5602:
5599:
5590:
5583:
5580:
5577:
5568:
5562:
5557:
5551:
5543:
5539:
5531:
5530:
5529:
5528:
5527:
5526:and defining
5511:
5507:
5486:
5459:
5455:
5451:
5448:
5445:
5442:
5435:
5434:
5433:
5432:
5431:
5417:
5410:by splitting
5397:
5394:
5391:
5383:
5367:
5361:
5358:
5353:
5349:
5328:
5308:
5288:
5268:
5245:
5242:
5239:
5229:
5228:inner product
5225:
5221:
5211:
5194:
5186:
5149:
5145:
5141:
5138:
5135:
5130:
5126:
5121:
5117:
5110:
5106:
5102:
5099:
5096:
5091:
5087:
5082:
5054:
5050:
5046:
5043:
5040:
5035:
5031:
5026:
5001:
4997:
4993:
4990:
4987:
4982:
4978:
4967:
4941:
4937:
4916:
4909:matrix whose
4896:
4893:
4890:
4866:
4862:
4858:
4855:
4852:
4847:
4843:
4835:
4831:
4827:
4824:
4821:
4816:
4812:
4801:
4772:
4767:
4764:
4752:
4748:
4744:
4741:
4738:
4733:
4729:
4721:
4717:
4713:
4710:
4707:
4702:
4698:
4687:
4673:
4669:
4665:
4662:
4659:
4654:
4650:
4639:
4632:
4625:
4621:
4617:
4614:
4611:
4606:
4602:
4591:
4580:
4579:
4578:
4577:
4576:
4558:
4554:
4550:
4547:
4544:
4539:
4535:
4530:
4505:
4501:
4497:
4494:
4491:
4486:
4482:
4477:
4451:
4444:
4440:
4436:
4433:
4430:
4425:
4421:
4416:
4408:
4404:
4400:
4397:
4394:
4389:
4385:
4374:
4367:
4360:
4356:
4352:
4349:
4346:
4341:
4337:
4332:
4324:
4320:
4316:
4313:
4310:
4305:
4301:
4290:
4279:
4278:
4277:
4276:
4275:
4257:
4253:
4249:
4246:
4243:
4238:
4234:
4229:
4204:
4200:
4196:
4193:
4190:
4185:
4181:
4176:
4151:
4147:
4143:
4140:
4137:
4132:
4128:
4123:
4119:
4112:
4108:
4104:
4101:
4098:
4093:
4089:
4084:
4059:
4055:
4051:
4048:
4045:
4040:
4036:
4031:
4005:
4001:
3997:
3994:
3991:
3986:
3982:
3958:
3955:
3952:
3925:
3922:
3910:
3906:
3902:
3899:
3896:
3891:
3887:
3882:
3875:
3872:
3865:
3861:
3857:
3854:
3851:
3846:
3842:
3831:
3820:
3819:
3818:
3817:
3816:
3802:
3799:
3793:
3790:
3787:
3764:
3740:
3736:
3732:
3729:
3726:
3721:
3717:
3712:
3703:
3683:
3679:
3675:
3672:
3669:
3664:
3660:
3655:
3630:
3626:
3622:
3619:
3616:
3611:
3607:
3602:
3590:representing
3573:
3569:
3565:
3562:
3559:
3554:
3550:
3545:
3521:
3513:
3500:
3497:
3474:
3451:
3431:
3428:
3425:
3400:
3396:
3392:
3389:
3386:
3381:
3377:
3351:
3348:
3341:
3337:
3333:
3330:
3327:
3322:
3318:
3313:
3309:
3285:
3281:
3277:
3274:
3271:
3266:
3262:
3257:
3236:
3233:
3227:
3224:
3221:
3194:
3190:
3186:
3183:
3180:
3175:
3171:
3166:
3145:
3123:
3119:
3098:
3074:
3070:
3066:
3063:
3060:
3055:
3051:
3046:
3025:
3022:
3019:
2999:
2976:
2968:
2955:
2952:
2928:
2924:
2920:
2917:
2914:
2909:
2905:
2900:
2879:
2876:
2871:
2867:
2863:
2860:
2857:
2852:
2848:
2844:
2841:
2832:
2818:
2810:
2806:
2790:
2765:
2761:
2757:
2754:
2751:
2746:
2742:
2714:
2710:
2706:
2703:
2700:
2695:
2691:
2686:
2665:
2662:
2659:
2639:
2636:
2631:
2627:
2623:
2620:
2617:
2612:
2608:
2604:
2601:
2581:
2561:
2541:
2521:
2496:
2493:
2489:
2478:with entries
2465:
2445:
2442:
2436:
2433:
2430:
2400:
2392:
2389:
2386:
2383:
2380:
2376:
2370:
2365:
2358:
2355:
2352:
2349:
2346:
2342:
2334:
2327:
2318:
2315:
2312:
2308:
2302:
2297:
2290:
2287:
2284:
2280:
2272:
2262:
2253:
2245:
2239:
2230:
2229:
2228:
2227:
2226:
2212:
2192:
2184:
2165:
2162:
2159:
2153:
2150:
2147:
2144:
2124:
2116:
2100:
2080:
2077:
2074:
2065:
2048:
2025:
2005:
1985:
1982:
1979:
1959:
1939:
1936:
1933:
1926:
1922:
1903:
1900:
1897:
1891:
1888:
1885:
1882:
1857:
1854:
1851:
1842:
1832:
1831:
1830:
1829:
1828:
1811:
1803:
1790:
1787:
1761:
1738:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1678:
1674:
1653:
1633:
1626:
1622:
1606:
1603:
1600:
1577:
1569:
1556:
1553:
1545:
1524:
1520:
1516:
1513:
1510:
1505:
1501:
1489:
1474:
1451:
1443:
1413:
1410:
1407:
1387:
1362:
1358:
1354:
1351:
1348:
1343:
1339:
1313:
1309:
1288:
1268:
1260:
1241:
1233:
1209:
1205:
1199:
1197:
1192:
1185:
1178:
1173:
1172:perpendicular
1167:
1161:
1157:
1138:
1135:
1132:
1118:
1108:
1104:
1097:
1086:
1084:
1080:
1076:
1072:
1056:
1036:
1028:
1023:
1021:
1005:
985:
965:
962:
959:
934:
919:
885:
878:assigning to
877:
859:
830:
803:
795:
794:tangent space
779:
776:
773:
751:
724:
717:of dimension
704:
696:
692:
687:
685:
681:
677:
673:
669:
659:
645:
625:
605:
582:
579:
576:
542:
534:
501:
498:
495:
461:
453:
434:
432:
428:
424:
403:
388:
370:
365:
363:
344:
341:
338:
332:
324:
321:
317:
313:
297:
277:
254:
239:
223:
200:
192:
162:
155:
139:
132:
129:-dimensional
116:
108:
105:
89:
81:
77:
58:
50:
33:
29:
22:
21290:
21266:
21227:
21224:Lee, John M.
21206:
21186:
21165:
21144:
21101:
21073:. Retrieved
21037:
21031:
21014:
20990:
20985:
20942:
20938:
20932:
20920:. Retrieved
20916:
20906:
20863:
20859:
20853:
20818:
20814:
20808:
20773:
20764:
20747:
20743:
20733:
20698:
20659:
20655:
20609:
20605:
20588:
20583:
20554:
20550:
20536:
20511:
20507:
20501:
20458:
20454:
20448:
20421:
20417:
20407:
20382:
20378:
20372:
20344:. New York:
20340:
20309:
20271:
20258:
20246:
20221:
20211:
20206:, pp. 57–59.
20199:
20187:
20175:
20135:
20043:
20024:
20017:
20005:KP hierarchy
19998:
19994:Bethe ansatz
19992:, using the
19990:Gaudin model
19975:
19972:
19969:Applications
19962:
19949:
19939:
19625:
19484:
19322:
19320:
19120:
19008:
18771:Haar measure
18769:be the unit
18660:
18594:
18487:
18338:
18130:
18122:
17929:
17625:
17619:
17502:
17409:
17398:
17042:
16981:
16907:
16673:
16666:
16354:
16162:
15971:
15776:
15677:
15342:
15120:
14617:
14280:
13950:
13282:
13275:
13091:
12774:
12569:
12426:
12106:
11898:
11809:
11742:
11581:homomorphism
11329:
11073:
10670:
10576:
10415:on a scheme
10386:
10366:
10354:
10350:
10346:
10339:
10126:
9879:
9876:
9643:
9404:
9022:
8949:non-singular
8909:, the group
8888:ground field
8799:
8772:
8658:
8621:
8348:
8160:transitively
8117:
7799:
7695:self-adjoint
7426:
7316:
6933:
6708:of matrices
6578:
6572:
6568:
6564:
6560:
6556:
6552:
6546:
6520:denotes the
6454:
6365:with metric
6363:metric space
6228:
5961:
5639:
5478:
5217:
4789:
4468:
3944:
2833:
2783:-th rows of
2419:
2066:
1874:
1623:rectangular
1215:
1203:
1200:
1190:
1183:
1176:
1165:
1162:
1160:dimensions.
1155:
1106:
1102:
1095:
1092:
1024:
737:embedded in
688:
665:
435:
366:
131:vector space
32:Grassmannian
31:
25:
21153:section 1.2
20922:17 December
20756:2433/102800
20253:, II.3.6.3.
19955:pure spinor
19207:, we have
19038:; that is,
18123:otherwise.
17613:into those
15972:Taking the
13546:with basis
13219:such that
13114:determinant
12427:is the set
9736:(the first
9644:If we take
9129:. The real
8775:left cosets
7611:satisfying
7542:of complex
4469:where both
3138:-th row of
2805:nonsingular
2514:determines
668:topological
104:dimensional
28:mathematics
21325:Categories
21132:0836.14001
21087:References
21075:2008-09-05
20945:(10): 30.
20821:: 83–151.
20770:Harnad, J.
20744:数理解析研究所講究録
20695:Harnad, J.
20302:0836.14001
20050:grand tour
19947:theory of
19863:for which
18628:zero-modes
18626:fermionic
18381:direct sum
18306:cohomology
18201:-plane in
18181:defines a
16908:of weight
16841:. For any
13480:, and let
12962:, and let
11901:surjective
11747:-schemes:
10875:, namely:
10853:dual space
10741:projective
8622:under the
8443:stabilizer
7800:where the
7752:has trace
6892:has trace
4956:th row of
3038:submatrix
2678:submatrix
1925:invertible
1083:isomorphic
672:continuous
662:Motivation
21254:808682771
21054:1618-1913
21007:full text
20952:1312.2007
20873:1106.0023
20828:0902.4433
20800:222379146
20750:: 30–46.
20725:222379146
20678:0034-5318
20634:0031-9015
20564:0711.4079
20493:123324468
20140:. In the
20072:Gauss map
19917:∈
19904:∀
19848:⊂
19781:⊂
19594:−
19585:
19579:×
19567:
19544:
19508:~
19443:~
19285:γ
19216:γ
19165:θ
19132:∈
19083:γ
19052:γ
18983:∈
18962:
18956:∈
18941:θ
18913:γ
18899:, define
18863:⊂
18816:∈
18751:θ
18601:instanton
18573:−
18559:−
18536:−
18312:, by the
18266:−
18035:χ
17943:χ
17895:χ
17876:χ
17863:−
17850:χ
17833:−
17819:−
17807:−
17800:χ
17781:χ
17747:−
17671:−
17650:−
17539:⊂
17506:. Fix a
17422:χ
17359:λ
17355:−
17343:−
17332:∩
17323:
17286:∈
17257:λ
17166:∈
17119:⊂
17099:λ
17061:−
17022:≥
17013:λ
17009:≥
17006:⋯
17003:≥
16994:λ
16959:λ
16938:∑
16926:λ
16884:λ
16877:⋯
16865:λ
16855:λ
16763:⊂
16760:⋯
16757:⊂
16744:⊂
16637:−
16589:∗
16530:−
16454:∗
16407:∗
16399:⊂
16366:⊂
16333:∗
16318:−
16300:↔
16249:∗
16218:−
16143:→
16138:∗
16130:→
16125:∗
16117:→
16112:∗
16090:→
16061:∗
16034:∗
16007:∗
15952:→
15938:→
15932:→
15926:→
15837:−
15808:⊂
15745:Λ
15686:ι
15612:−
15457:Λ
15445:⊂
15415:ι
15354:
15319:Λ
15289:ι
15219:…
15170:…
15161:^
15142:…
15077:…
15068:^
15049:…
15008:−
14994:…
14971:ℓ
14960:−
14931:∑
14834:ι
14822:, or the
14777:−
14751:≤
14729:⋯
14703:≤
14677:≤
14669:−
14655:⋯
14629:≤
14591:Λ
14505:⋯
14460:Λ
14410:ι
14354:⋯
14294:∈
14261:≤
14245:⋯
14229:≤
14203:…
14139:…
14093:×
14055:…
13994:≤
13978:⋯
13962:≤
13908:⋯
13886:⋮
13881:⋱
13876:⋮
13854:⋯
13815:⋯
13727:…
13625:⋯
13570:⋯
13491:⊂
13432:⋯
13344:Λ
13238:ι
13235:∧
13204:∈
13168:ι
13128:ι
13073:⋅
13033:∧
13030:⋯
13027:∧
12999:ι
12970:ι
12914:⋯
12869:ι
12786:⊂
12742:Λ
12728:→
12699:ι
12646:Λ
12531:∈
12525:∣
12486:×
12460:∈
12378:×
12261:−
12081:×
11870:×
11848:→
11763:→
11690:×
11614:→
11290:over the
11250:⊗
11151:→
11018:≃
11001:×
10537:⊗
10311:−
10299:×
10195:⊂
10097:⊥
10069:⊥
10050:×
9848:−
9836:×
9712:⊂
9616:⊥
9593:⊥
9579:×
9423:⊥
9382:⊥
9354:⊥
9335:×
9265:⊂
8880:Lie group
8656:-action.
8590:⊂
8528:∈
8423:∈
8360:∈
8266:∈
8220:⊂
8091:†
8051:∑
7996:⋯
7948:⊂
7862:⋯
7826:⟩
7822:⋅
7816:⋅
7812:⟨
7710:†
7579:∈
7568:matrices
7553:×
7510:⊂
7380:⊂
7341:⋯
7255:∑
7184:∈
7086:∈
6903:
6835:symmetric
6719:∈
6676:⊂
6627:×
6477:⊂
6435:‖
6417:−
6404:‖
6319:Hausdorff
6288:†
6266:∣
6243:∈
6209:×
6174:×
6156:matrices
6141:×
5908:†
5881:∣
5858:∈
5847:∼
5611:⊥
5603:∈
5581:∈
5512:⊥
5460:⊥
5452:⊕
5395:⊂
5365:→
5249:⟩
5246:⋅
5240:⋅
5237:⟨
5139:…
5100:…
5044:…
4991:…
4971:^
4894:×
4856:…
4825:…
4805:^
4765:−
4742:…
4711:…
4691:^
4663:…
4643:^
4615:…
4595:^
4548:…
4495:…
4434:…
4398:…
4378:^
4350:…
4314:…
4294:^
4247:…
4194:…
4141:…
4120:∩
4102:…
4049:…
3995:…
3956:×
3923:−
3900:…
3855:…
3835:^
3800:×
3791:−
3730:…
3673:…
3620:…
3563:…
3501:∈
3429:×
3390:…
3349:−
3331:…
3275:…
3234:×
3225:−
3184:…
3064:…
3023:×
2956:∈
2918:…
2877:≤
2861:⋯
2845:≤
2755:…
2704:…
2663:×
2637:≤
2621:⋯
2605:≤
2443:×
2434:−
2384:−
2371:⋯
2366:⋯
2350:−
2335:⋮
2328:⋮
2303:⋯
2298:⋯
2263:⋱
2148:∈
2078:×
1998:matrices
1983:×
1937:×
1886:∈
1846:~
1791:∈
1765:~
1713:…
1604:×
1557:∈
1514:…
1411:⊂
1352:…
1216:To endow
1136:−
1079:homotopic
1020:Gauss map
963:∈
777:∈
342:−
21288:(2013).
21264:(1995).
21226:(2012).
21164:(1974).
21142:(2003).
21100:(1994).
21070:14420180
21023:(1999).
21003:abstract
20898:51759294
20845:18390193
20545:(2009).
20270:(1994),
20219:(1971).
20180:Lee 2012
20130:K-theory
20056:See also
19996:method.
19945:Cartan's
19323:oriented
19156:. Since
19121:for all
18805:and fix
18097:⌋
18084:⌊
18075:⌋
18062:⌊
11051:′
11014:′
10947:′
10935:-scheme
10695:-scheme
10475:-scheme
10127:or, for
8960:complete
8389:, where
6518:‖
6514:‖
6473:′
6429:′
6394:′
5660:of rank
3777:-valued
2420:and the
2205:rows of
2018:of rank
1619:maximal
1073:must be
1071:theorems
429:below.)
21124:1288523
21062:1813224
20977:7717260
20957:Bibcode
20878:Bibcode
20614:Bibcode
20528:2373211
20485:2372896
20426:Bibcode
20387:Bibcode
20364:0631850
20294:1288523
20020:soliton
19950:spinors
19646:on the
18773:on the
18653:units.
18427:of the
16671:called
15773:Duality
13368:of the
13189:as the
13116:of the
10673:functor
10342:compact
9247:-space
8441:of the
8162:on the
7124:to the
6361:into a
6316:compact
3249:matrix
1919:of the
1542:. The
425:. (See
320:compact
316:complex
290:. When
152:over a
78:) is a
21310:
21274:
21252:
21242:
21213:
21194:
21172:
21130:
21122:
21112:
21068:
21060:
21052:
20997:
20975:
20896:
20843:
20798:
20788:
20723:
20713:
20676:
20632:
20526:
20491:
20483:
20362:
20352:
20300:
20292:
20282:
20233:
20080:, see
20070:, see
18681:is an
18599:of an
17202:whose
17043:whose
15777:Every
15381:, and
15121:where
13703:, and
13598:. Let
13061:where
10373:scheme
9903:, the
9405:where
8943:is an
7317:where
6563:) and
5380:whose
5071:, and
4790:where
3091:whose
3012:, the
2892:, let
1646:whose
1625:matrix
1206:(2, 4)
1188:, and
1186:(1, 3)
1179:(2, 3)
1075:proved
874:. The
792:, the
680:closed
109:of an
30:, the
21149:(PDF)
21066:S2CID
21028:(PDF)
20973:S2CID
20947:arXiv
20894:S2CID
20868:arXiv
20841:S2CID
20823:arXiv
20796:S2CID
20721:S2CID
20559:arXiv
20524:JSTOR
20489:S2CID
20481:JSTOR
20316:arXiv
20168:Notes
19309:is a
18661:When
18603:with
18248:of a
16576:with
16441:with
15343:When
13506:be a
13160:from
12801:is a
11535:over
10671:This
10411:be a
9877:Over
9023:Over
8982:is a
7634:is a
6774:is a
5729:trace
5382:image
3704:from
1301:with
1259:basis
1194:(the
310:is a
154:field
21308:ISBN
21272:ISBN
21250:OCLC
21240:ISBN
21211:ISBN
21192:ISBN
21170:ISBN
21110:ISBN
21050:ISSN
20995:ISBN
20943:2014
20924:2020
20786:ISBN
20711:ISBN
20674:ISSN
20630:ISSN
20350:ISBN
20280:ISBN
20231:ISBN
20084:and
20025:The
18488:The
18403:and
18359:and
18310:ring
18021:and
17997:and
17995:even
16509:and
16204:and
15974:dual
14792:and
14732:<
14716:<
14692:and
14658:<
14642:<
14248:<
14242:<
13981:<
13975:<
13755:the
13460:for
13276:The
13191:span
12942:for
12570:The
12153:and
11135:Spec
10387:Let
10162:and
9679:and
9280:is
8844:and
8157:acts
6579:Let
5811::
4522:and
4221:and
2864:<
2858:<
2624:<
2618:<
1751:and
1621:rank
1261:for
1163:For
1105:(1,
1093:For
678:and
676:open
312:real
21300:doi
21128:Zbl
21042:doi
20965:doi
20886:doi
20864:198
20833:doi
20819:123
20778:doi
20752:hdl
20748:439
20703:doi
20664:doi
20622:doi
20569:doi
20516:doi
20471:hdl
20463:doi
20434:doi
20395:doi
20298:Zbl
20251:EGA
20128:in
20107:or
20076:In
20029:of
18316:of
18019:odd
18017:is
17993:is
17973:if
17496:of
17320:dim
16021:in
15886:of
15351:dim
14876:of
14766:of
14533:of
14431:of
14382:of
14009:of
12667:of
12634:th
12351:in
12276:in
12054:in
11460:of
11094:of
10855:of
10743:if
10597:on
10499:of
9464:in
9109:on
9045:or
8986:of
8822:or
8777:of
8773:of
8445:of
8024:is
7693:is
7026:of
6833:is
6492:of
6191:,
5997:or
5479:of
5384:is
5261:on
3464:in
2803:is
1923:of
1168:= 2
1158:− 1
1153:of
1098:= 1
902:to
876:map
796:to
314:or
26:In
21327::
21306:.
21298:.
21294:.
21248:.
21238:.
21230:.
21160:;
21126:.
21120:MR
21118:.
21096:;
21064:.
21058:MR
21056:.
21048:.
21038:90
21036:.
21030:.
21005:,
20971:.
20963:.
20955:.
20941:.
20915:.
20892:.
20884:.
20876:.
20862:.
20839:.
20831:.
20817:.
20794:.
20784:.
20746:.
20742:.
20719:.
20709:.
20686:^
20672:.
20660:19
20658:.
20654:.
20642:^
20628:.
20620:.
20610:50
20608:.
20596:^
20567:.
20555:22
20553:.
20549:.
20522:.
20512:85
20510:.
20487:.
20479:.
20469:.
20459:83
20457:.
20432:.
20422:36
20420:.
20416:.
20393:.
20383:33
20381:.
20360:MR
20358:.
20348:.
20330:^
20296:,
20290:MR
20288:,
20266:;
20229:.
20052:.
20041:.
19965:.
19582:SO
19564:SO
19541:SO
19482:.
18471:1.
17913:1.
17407:.
16659:.
16146:0.
15955:0.
15910::
15661:0.
15653:23
15643:14
15630:24
15620:13
15607:34
15597:12
15562:34
15549:24
15536:23
15523:14
15510:13
15497:12
15340:.
14615:.
13847:11
12687:.
11322:.
10668:.
10526::=
10379:.
10359:.
10353:−
10219:,
9020:.
8797:.
8519::=
8486::=
8472::
8047::=
7638::
7251::=
6900:tr
6837::
6778::
6565:Gr
6553:Gr
6401::=
6081:,
5698::=
3873::=
2064:.
1693:,
1208:.
1204:Gr
1184:Gr
1181:,
1177:Gr
1103:Gr
658:.
558:,
477:,
364:.
21316:.
21302::
21280:.
21256:.
21200:.
21178:.
21134:.
21078:.
21044::
21009:)
20979:.
20967::
20959::
20949::
20926:.
20900:.
20888::
20880::
20870::
20847:.
20835::
20825::
20802:.
20780::
20758:.
20754::
20727:.
20705::
20680:.
20666::
20636:.
20624::
20616::
20577:.
20571::
20561::
20530:.
20518::
20495:.
20473::
20465::
20442:.
20436::
20428::
20401:.
20397::
20389::
20366:.
20324:.
20318::
20239:.
20148:.
20138:)
20136:n
20111:.
20088:.
19923:.
19920:w
19914:v
19911:,
19908:u
19900:,
19897:0
19894:=
19891:)
19888:v
19885:,
19882:u
19879:(
19876:Q
19851:V
19845:w
19825:k
19805:)
19802:V
19799:(
19794:k
19789:r
19786:G
19778:)
19775:Q
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19769:V
19766:(
19761:0
19756:k
19751:r
19748:G
19726:)
19723:Q
19720:,
19717:V
19714:(
19709:0
19704:k
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19696:G
19674:V
19654:n
19634:Q
19606:.
19603:)
19600:)
19597:k
19591:n
19588:(
19576:)
19573:k
19570:(
19561:(
19557:/
19553:)
19550:n
19547:(
19538:=
19535:)
19530:n
19525:R
19520:(
19515:k
19504:r
19501:G
19470:)
19465:n
19460:R
19455:(
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19436:G
19411:)
19406:n
19401:R
19396:(
19391:k
19386:r
19383:G
19359:n
19354:R
19332:k
19295:n
19292:,
19289:k
19264:1
19261:=
19258:)
19255:)
19252:V
19249:(
19244:k
19239:r
19236:G
19231:(
19226:n
19223:,
19220:k
19195:1
19192:=
19189:)
19186:)
19183:n
19180:(
19177:O
19174:(
19169:n
19144:)
19141:n
19138:(
19135:O
19129:g
19104:)
19101:A
19098:(
19093:n
19090:,
19087:k
19079:=
19076:)
19073:A
19070:g
19067:(
19062:n
19059:,
19056:k
19026:)
19023:n
19020:(
19017:O
18992:.
18989:}
18986:A
18980:w
18977:g
18974::
18971:)
18968:n
18965:(
18959:O
18953:g
18950:{
18945:n
18937:=
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7656:=
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