1544:
1300:
1777:
217:— Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields. See
189:— Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to infinite groups by replacement of the average with an integral, provided that an acceptable notion of integral can be defined. This can be done for locally compact groups, using the
1354:
248:. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different (and much less well understood). The analytic techniques used for studying Lie groups must be replaced by techniques from
1110:
1591:
27:
138:
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are:
1539:{\displaystyle \sigma \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \sigma \left(u\right)={\begin{bmatrix}u&0\\0&1\\\end{bmatrix}}\qquad \sigma \left(u^{2}\right)={\begin{bmatrix}u^{2}&0\\0&1\\\end{bmatrix}}.}
1295:{\displaystyle \rho \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \rho \left(u\right)={\begin{bmatrix}1&0\\0&u\\\end{bmatrix}}\qquad \rho \left(u^{2}\right)={\begin{bmatrix}1&0\\0&u^{2}\\\end{bmatrix}}.}
572:
1882:
1772:{\displaystyle \tau \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \tau \left(u\right)={\begin{bmatrix}a&-b\\b&a\\\end{bmatrix}}\qquad \tau \left(u^{2}\right)={\begin{bmatrix}a&b\\-b&a\\\end{bmatrix}}}
2768:
2194:
In the example above, the first two representations given (ρ and σ) are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}), while the third representation (τ) is irreducible.
292:
on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a
852:
1036:
438:
2436:
262:— The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The
1964:
1583:
1346:
1102:
110:
is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a
449:
2321:
2106:
2074:
1788:
660:
1993:
2042:
2459:
2013:
1908:
2857:
130:
for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.
148:— Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to
780:
2156:. The representation of dimension zero is considered to be neither reducible nor irreducible, just as the number 1 is considered to be neither
972:
2884:
395:
382:
2610:, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of
84:
In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
2743:
174:
2825:
332:
of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the
2738:
2327:
2733:
222:
2879:
2797:
2644:
745:
2809:
218:
169:
2713:
1913:
2801:
2630:
2507:
2131:
2117:
281:
1559:
1322:
1078:
2748:
2674:
2603:
2191:, since the characteristic of the complex numbers is zero, which never divides the size of a group.
567:{\displaystyle \rho (g_{1}g_{2})=\rho (g_{1})\rho (g_{2}),\qquad {\text{for all }}g_{1},g_{2}\in G.}
2168:
686:
329:
157:
1877:{\displaystyle a={\text{Re}}(u)=-{\tfrac {1}{2}},\qquad b={\text{Im}}(u)={\tfrac {\sqrt {3}}{2}}.}
861:
20:
2861:. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
2282:
2523:
2231:
2079:
2047:
639:
613:
229:
206:
2152:; if it has a proper subrepresentation of nonzero dimension, the representation is said to be
270:
Lie groups cannot be classified in the same way. The general theory for Lie groups deals with
2874:
2688:
185:
78:
62:
47:
2835:
2184:
2180:
1971:
333:
325:
2021:
8:
2217:
363:
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305:
153:
55:
31:
240:) — These are the analogues of Lie groups, but over more general fields than just
2728:
2579:
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2444:
1998:
1910:
be the space of homogeneous degree-3 polynomials over the complex numbers in variables
1893:
370:
271:
249:
198:
115:
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2821:
2224:
2136:
678:
662:
202:
194:
74:
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2678:
2615:
2157:
253:
2831:
2718:
2700:
2696:
2490:
2188:
149:
308:
over which the vector space is defined. The most important case is the field of
103:
of a physical system affects the solutions of equations describing that system.
309:
100:
92:
35:
2817:
2187:). This holds in particular for any representation of a finite group over the
601:
itself as the representation when the homomorphism is clear from the context.
2868:
2843:
2660:
2144:
has exactly two subrepresentations, namely the zero-dimensional subspace and
321:
294:
276:
181:
2670:
Two types of representations closely related to linear representations are:
847:{\displaystyle \ker \rho =\left\{g\in G\mid \rho (g)=\mathrm {id} \right\}.}
2692:
2176:
2161:
1072:
356:
317:
313:
298:
289:
236:
190:
144:
111:
88:
70:
66:
2474:
939:
43:
2849:
2485:. Thus we may equivalently define a permutation representation to be a
213:
26:
2812:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
2470:
1306:
876:
59:
328:
fields are easier to handle than non-algebraically closed ones. The
266:
have a deep theory, building on the compact case. The complementary
16:
Group homomorphism into the general linear group over a vector space
2634:
2527:
73:); in particular, they can be used to represent group elements as
2551:
96:
38:, consisting of reflections and rotations, transform the polygon.
1031:{\displaystyle \alpha \circ \rho (g)\circ \alpha ^{-1}=\pi (g).}
2858:
201:
describes the theory for commutative groups, as a generalised
2682:
879:; in other words, one whose kernel is the trivial subgroup {
433:{\displaystyle \rho \colon G\to \mathrm {GL} \left(V\right)}
1075:
under multiplication. This group has a representation ρ on
2175:
does not divide the size of the group, representations of
288:
Representation theory also depends heavily on the type of
133:
2848:. Introduction to representation theory with emphasis on
2465:. This condition and the axioms for a group imply that ρ(
597:
of the representation. It is common practice to refer to
118:
of an object. If the object is a vector space we have a
173:; this special case has very different properties. See
2513:
2506:
For more information on this topic see the article on
1858:
1819:
1735:
1671:
1617:
1495:
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1380:
1251:
1190:
1136:
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2001:
1974:
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1562:
1357:
1325:
1113:
1081:
975:
783:
642:
452:
398:
274:
of the two types, by means of general results called
2681:. These can be described as "linear representations
167:
divides the order of the group, then this is called
2453:
2430:
2315:
2203:
2100:
2068:
2036:
2007:
1987:
1958:
1902:
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1340:
1294:
1096:
1030:
883:} consisting only of the group's identity element.
846:
654:
566:
432:
77:so that the group operation can be represented by
1348:, isomorphic to the previous one, is σ given by:
34:"acts" on an object. A simple example is how the
2866:
2431:{\displaystyle \rho (g_{1}g_{2})=\rho (g_{1})],}
1305:This representation is faithful because ρ is a
2148:itself, then the representation is said to be
312:. The other important cases are the field of
2796:
193:. The resulting theory is a central part of
126:for the general notion and reserve the term
2530:in this category are just the elements of
1565:
1328:
1084:
25:
2183:of irreducible subrepresentations (see
2015:by permutation of the three variables.
389:. That is, a representation is a map
134:Branches of group representation theory
2867:
2744:Representation theory of finite groups
1556:may also be faithfully represented on
175:Representation theory of finite groups
91:problems to be reduced to problems in
2806:Representation theory. A first course
763:is defined as the normal subgroup of
87:Representations of groups allow many
2739:List of representation theory topics
256:causes many technical complications.
2562:. Such a functor selects an object
2514:Representations in other categories
304:One must also consider the type of
156:of scalars of the vector space has
13:
2633:, the objects obtained are called
2198:
1959:{\displaystyle x_{1},x_{2},x_{3}.}
832:
829:
415:
412:
14:
2896:
2643:For another example consider the
2212:(also known as a group action or
864:is one in which the homomorphism
2734:List of harmonic analysis topics
1578:{\displaystyle \mathbb {R} ^{2}}
1341:{\displaystyle \mathbb {C} ^{2}}
1097:{\displaystyle \mathbb {C} ^{2}}
2885:Representation theory of groups
2204:Set-theoretical representations
2111:
1833:
1705:
1648:
1465:
1411:
1221:
1167:
938:if there exists a vector space
771:is the identity transformation:
526:
280:, which is a generalization of
223:Representations of Lie algebras
36:symmetries of a regular polygon
2761:
2645:category of topological spaces
2570:and a group homomorphism from
2534:. Given an arbitrary category
2422:
2419:
2413:
2410:
2397:
2391:
2388:
2375:
2366:
2360:
2357:
2334:
2304:
2298:
2295:
2289:
2167:Under the assumption that the
2031:
2025:
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1803:
1022:
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991:
985:
822:
816:
520:
507:
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488:
479:
456:
408:
339:
260:Non-compact topological groups
1:
2810:Graduate Texts in Mathematics
2790:
2663:group of a topological space
219:Representations of Lie groups
170:modular representation theory
69:to itself (i.e. vector space
2242:, the set of functions from
2210:set-theoretic representation
2130:that is invariant under the
1048:Consider the complex number
252:, where the relatively weak
7:
2714:Irreducible representations
2707:
2461:is the identity element of
1312:Another representation for
1052:= e which has the property
1043:
10:
2901:
2675:projective representations
2631:category of abelian groups
2316:{\displaystyle \rho (1)=x}
2214:permutation representation
2118:Irreducible representation
2115:
705:such that the application
18:
2818:10.1007/978-1-4612-0979-9
2749:Semisimple representation
2604:category of vector spaces
2179:can be decomposed into a
2101:{\displaystyle x_{2}^{3}}
2069:{\displaystyle x_{1}^{3}}
691:continuous representation
655:{\displaystyle n\times n}
612:it is common to choose a
108:representation of a group
2754:
2685:scalar transformations".
687:topological vector space
152:and to geometry. If the
99:, they describe how the
19:Not to be confused with
2655:are homomorphisms from
862:faithful representation
608:is of finite dimension
282:Wigner's classification
230:Linear algebraic groups
21:Presentation of a group
2769:"1.4: Representations"
2689:affine representations
2526:with a single object;
2455:
2432:
2317:
2102:
2070:
2038:
2009:
1989:
1960:
1904:
1878:
1773:
1579:
1540:
1342:
1296:
1098:
1032:
900:, two representations
848:
656:
568:
434:
186:locally compact groups
114:from the group to the
63:linear transformations
39:
30:A representation of a
2880:Representation theory
2691:: in the category of
2677:: in the category of
2651:. Representations in
2456:
2433:
2318:
2103:
2071:
2039:
2010:
1990:
1988:{\displaystyle S_{3}}
1961:
1905:
1879:
1774:
1580:
1541:
1343:
1297:
1099:
1033:
849:
657:
585:and the dimension of
569:
435:
264:semisimple Lie groups
120:linear representation
79:matrix multiplication
52:group representations
48:representation theory
29:
2773:Chemistry LibreTexts
2445:
2328:
2283:
2250:, such that for all
2080:
2048:
2037:{\displaystyle (12)}
2022:
1999:
1972:
1914:
1894:
1789:
1592:
1560:
1355:
1323:
1111:
1079:
973:
781:
755:of a representation
701:is a representation
640:
583:representation space
450:
396:
383:general linear group
326:algebraically closed
233:(or more generally
2699:acts affinely upon
2695:. For example, the
2522:can be viewed as a
2097:
2065:
663:invertible matrices
272:semidirect products
75:invertible matrices
2855:Yurii I. Lyubich.
2729:Molecular symmetry
2589:In the case where
2580:automorphism group
2487:group homomorphism
2451:
2428:
2313:
2098:
2083:
2066:
2051:
2034:
2005:
1985:
1956:
1900:
1874:
1869:
1828:
1769:
1763:
1699:
1642:
1575:
1536:
1527:
1459:
1405:
1338:
1292:
1283:
1215:
1161:
1094:
1028:
844:
767:whose image under
652:
604:In the case where
564:
430:
371:group homomorphism
334:order of the group
250:algebraic geometry
207:Peter–Weyl theorem
199:Pontryagin duality
122:. Some people use
116:automorphism group
54:describe abstract
40:
2827:978-0-387-97495-8
2679:projective spaces
2454:{\displaystyle 1}
2185:Maschke's theorem
2137:subrepresentation
2008:{\displaystyle V}
1903:{\displaystyle V}
1887:Another example:
1868:
1864:
1843:
1827:
1801:
679:topological group
530:
203:Fourier transform
195:harmonic analysis
2892:
2847:
2784:
2783:
2781:
2780:
2765:
2724:Character theory
2616:category of sets
2460:
2458:
2457:
2452:
2437:
2435:
2434:
2429:
2409:
2408:
2387:
2386:
2356:
2355:
2346:
2345:
2322:
2320:
2319:
2314:
2107:
2105:
2104:
2099:
2096:
2091:
2075:
2073:
2072:
2067:
2064:
2059:
2043:
2041:
2040:
2035:
2014:
2012:
2011:
2006:
1994:
1992:
1991:
1986:
1984:
1983:
1965:
1963:
1962:
1957:
1952:
1951:
1939:
1938:
1926:
1925:
1909:
1907:
1906:
1901:
1883:
1881:
1880:
1875:
1870:
1860:
1859:
1844:
1841:
1829:
1820:
1802:
1799:
1778:
1776:
1775:
1770:
1768:
1767:
1726:
1722:
1721:
1704:
1703:
1662:
1647:
1646:
1608:
1584:
1582:
1581:
1576:
1574:
1573:
1568:
1545:
1543:
1542:
1537:
1532:
1531:
1507:
1506:
1486:
1482:
1481:
1464:
1463:
1425:
1410:
1409:
1371:
1347:
1345:
1344:
1339:
1337:
1336:
1331:
1301:
1299:
1298:
1293:
1288:
1287:
1280:
1279:
1242:
1238:
1237:
1220:
1219:
1181:
1166:
1165:
1127:
1103:
1101:
1100:
1095:
1093:
1092:
1087:
1037:
1035:
1034:
1029:
1009:
1008:
955:so that for all
954:
929:
914:
874:
853:
851:
850:
845:
840:
836:
835:
743:
719:
661:
659:
658:
653:
635:
620:and identify GL(
573:
571:
570:
565:
554:
553:
541:
540:
531:
528:
519:
518:
500:
499:
478:
477:
468:
467:
439:
437:
436:
431:
429:
418:
320:, and fields of
254:Zariski topology
2900:
2899:
2895:
2894:
2893:
2891:
2890:
2889:
2865:
2864:
2828:
2798:Fulton, William
2793:
2788:
2787:
2778:
2776:
2767:
2766:
2762:
2757:
2719:Character table
2710:
2701:Euclidean space
2697:Euclidean group
2601:
2516:
2498:
2491:symmetric group
2446:
2443:
2442:
2404:
2400:
2382:
2378:
2351:
2347:
2341:
2337:
2329:
2326:
2325:
2284:
2281:
2280:
2263:
2256:
2206:
2201:
2199:Generalizations
2189:complex numbers
2120:
2114:
2092:
2087:
2081:
2078:
2077:
2060:
2055:
2049:
2046:
2045:
2023:
2020:
2019:
2000:
1997:
1996:
1979:
1975:
1973:
1970:
1969:
1947:
1943:
1934:
1930:
1921:
1917:
1915:
1912:
1911:
1895:
1892:
1891:
1857:
1840:
1818:
1798:
1790:
1787:
1786:
1762:
1761:
1756:
1747:
1746:
1741:
1731:
1730:
1717:
1713:
1709:
1698:
1697:
1692:
1686:
1685:
1677:
1667:
1666:
1652:
1641:
1640:
1635:
1629:
1628:
1623:
1613:
1612:
1598:
1593:
1590:
1589:
1585:by τ given by:
1569:
1564:
1563:
1561:
1558:
1557:
1555:
1526:
1525:
1520:
1514:
1513:
1508:
1502:
1498:
1491:
1490:
1477:
1473:
1469:
1458:
1457:
1452:
1446:
1445:
1440:
1430:
1429:
1415:
1404:
1403:
1398:
1392:
1391:
1386:
1376:
1375:
1361:
1356:
1353:
1352:
1332:
1327:
1326:
1324:
1321:
1320:
1318:
1282:
1281:
1275:
1271:
1269:
1263:
1262:
1257:
1247:
1246:
1233:
1229:
1225:
1214:
1213:
1208:
1202:
1201:
1196:
1186:
1185:
1171:
1160:
1159:
1154:
1148:
1147:
1142:
1132:
1131:
1117:
1112:
1109:
1108:
1088:
1083:
1082:
1080:
1077:
1076:
1062:
1046:
1001:
997:
974:
971:
970:
942:
930:are said to be
916:
901:
865:
828:
800:
796:
782:
779:
778:
721:
706:
641:
638:
637:
636:, the group of
625:
549:
545:
536:
532:
527:
514:
510:
495:
491:
473:
469:
463:
459:
451:
448:
447:
419:
411:
397:
394:
393:
342:
310:complex numbers
150:crystallography
136:
89:group-theoretic
24:
17:
12:
11:
5:
2898:
2888:
2887:
2882:
2877:
2863:
2862:
2853:
2826:
2792:
2789:
2786:
2785:
2759:
2758:
2756:
2753:
2752:
2751:
2746:
2741:
2736:
2731:
2726:
2721:
2716:
2709:
2706:
2705:
2704:
2686:
2597:
2540:representation
2515:
2512:
2494:
2489:from G to the
2450:
2439:
2438:
2427:
2424:
2421:
2418:
2415:
2412:
2407:
2403:
2399:
2396:
2393:
2390:
2385:
2381:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2354:
2350:
2344:
2340:
2336:
2333:
2323:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2261:
2254:
2230:is given by a
2205:
2202:
2200:
2197:
2169:characteristic
2116:Main article:
2113:
2110:
2095:
2090:
2086:
2063:
2058:
2054:
2033:
2030:
2027:
2018:For instance,
2004:
1982:
1978:
1955:
1950:
1946:
1942:
1937:
1933:
1929:
1924:
1920:
1899:
1885:
1884:
1873:
1867:
1863:
1856:
1853:
1850:
1847:
1839:
1836:
1832:
1826:
1823:
1817:
1814:
1811:
1808:
1805:
1797:
1794:
1780:
1779:
1766:
1760:
1757:
1755:
1752:
1749:
1748:
1745:
1742:
1740:
1737:
1736:
1734:
1729:
1725:
1720:
1716:
1712:
1708:
1702:
1696:
1693:
1691:
1688:
1687:
1684:
1681:
1678:
1676:
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892:vector spaces
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581:is called the
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346:representation
341:
338:
330:characteristic
324:. In general,
322:p-adic numbers
286:
285:
257:
226:
210:
182:Compact groups
178:
158:characteristic
135:
132:
128:representation
101:symmetry group
93:linear algebra
15:
9:
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2661:homeomorphism
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2177:finite groups
2174:
2171:of the field
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2159:
2155:
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2147:
2143:
2139:
2138:
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1118:
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1107:
1106:
1105:
1089:
1074:
1070:
1066:
1059:
1056:= 1. The set
1055:
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1019:
1013:
1010:
1005:
1002:
998:
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988:
982:
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714:
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704:
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692:
688:
684:
680:
676:
672:
671:
670:
668:
665:on the field
664:
649:
646:
643:
633:
629:
623:
619:
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611:
607:
602:
600:
596:
592:
588:
584:
580:
561:
558:
555:
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546:
542:
537:
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529:for all
523:
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337:
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331:
327:
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319:
318:finite fields
315:
311:
307:
302:
300:
296:
295:Hilbert space
291:
283:
279:
278:
277:Mackey theory
273:
269:
265:
261:
258:
255:
251:
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243:
239:
238:
237:group schemes
232:
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159:
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147:
146:
145:Finite groups
142:
141:
140:
131:
129:
125:
121:
117:
113:
109:
104:
102:
98:
94:
90:
85:
82:
80:
76:
72:
71:automorphisms
68:
64:
61:
57:
53:
49:
45:
37:
33:
28:
22:
2875:Group theory
2856:
2805:
2777:. Retrieved
2775:. 2019-09-04
2772:
2763:
2669:
2664:
2656:
2652:
2648:
2642:
2635:
2626:
2622:
2620:
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2607:
2598:
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2583:
2575:
2571:
2567:
2563:
2559:
2555:
2547:
2543:
2539:
2535:
2531:
2519:
2518:Every group
2517:
2508:group action
2505:
2500:
2495:
2482:
2478:
2466:
2462:
2440:
2273:
2269:
2265:
2258:
2251:
2247:
2243:
2239:
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2220:
2213:
2209:
2207:
2193:
2172:
2166:
2153:
2149:
2145:
2141:
2135:
2134:is called a
2132:group action
2127:
2123:
2121:
2112:Reducibility
2017:
1967:
1889:
1886:
1781:
1550:
1548:
1313:
1311:
1304:
1073:cyclic group
1068:
1064:
1057:
1053:
1049:
1047:
960:
956:
951:
947:
943:
935:
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690:
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594:
590:
586:
582:
578:
576:
442:
386:
378:
374:
366:
359:
357:vector space
352:
345:
343:
314:real numbers
303:
299:Banach space
290:vector space
287:
275:
267:
263:
259:
245:
241:
234:
228:
212:
205:. See also:
191:Haar measure
180:
168:
164:
160:
143:
137:
127:
123:
119:
112:homomorphism
107:
105:
86:
83:
67:vector space
58:in terms of
51:
44:mathematical
41:
2802:Harris, Joe
2475:permutation
2150:irreducible
2122:A subspace
940:isomorphism
759:of a group
720:defined by
340:Definitions
124:realization
2869:Categories
2850:Lie groups
2791:References
2779:2021-06-23
2477:) for all
2181:direct sum
1549:The group
1104:given by:
1071:} forms a
936:isomorphic
932:equivalent
888:Given two
746:continuous
443:such that
214:Lie groups
2844:246650103
2528:morphisms
2471:bijection
2395:ρ
2373:ρ
2332:ρ
2287:ρ
2234:ρ :
2158:composite
2154:reducible
1816:−
1751:−
1707:τ
1680:−
1650:τ
1596:τ
1467:σ
1413:σ
1359:σ
1223:ρ
1169:ρ
1115:ρ
1014:π
1003:−
999:α
995:∘
983:ρ
980:∘
977:α
877:injective
814:ρ
811:∣
805:∈
791:ρ
788:
707:Φ :
647:×
591:dimension
556:∈
505:ρ
486:ρ
454:ρ
409:→
403::
400:ρ
301:, etc.).
163:, and if
106:The term
60:bijective
46:field of
2804:(1991).
2708:See also
2638:-modules
2524:category
2268:and all
2232:function
1995:acts on
1044:Examples
946: :
920: :
905: :
284:methods.
268:solvable
2836:1153249
2659:to the
2614:in the
2578:), the
2574:to Aut(
2552:functor
2469:) is a
2216:) of a
624:) with
381:), the
362:over a
235:affine
97:physics
42:In the
2842:
2834:
2824:
2629:, the
2602:, the
2441:where
2044:sends
1782:where
1063:= {1,
753:kernel
595:degree
377:to GL(
197:. The
56:groups
2755:Notes
2683:up to
2621:When
2554:from
2550:is a
2223:on a
2218:group
2162:prime
2140:. If
1968:Then
924:→ GL(
909:→ GL(
869:→ GL(
685:is a
677:is a
614:basis
577:Here
373:from
369:is a
364:field
355:on a
350:group
348:of a
306:field
154:field
95:. In
65:of a
32:group
2840:OCLC
2822:ISBN
2595:Vect
2538:, a
2473:(or
2160:nor
1890:Let
915:and
896:and
751:The
730:) =
689:, a
681:and
616:for
221:and
2814:doi
2653:Top
2649:Top
2625:is
2593:is
2582:of
2566:in
2558:to
2546:in
2542:of
2499:of
2481:in
2272:in
2264:in
2246:to
2225:set
2126:of
2076:to
1319:on
959:in
934:or
875:is
785:ker
744:is
697:on
693:of
673:If
626:GL(
593:or
385:on
244:or
184:or
2871::
2838:.
2832:MR
2830:.
2820:.
2808:.
2800:;
2771:.
2667:.
2647:,
2640:.
2627:Ab
2618:.
2586:.
2510:.
2503:.
2276::
2257:,
2238:→
2208:A
2164:.
2108:.
2029:12
1842:Im
1800:Re
1309:.
1067:,
950:→
860:A
738:)(
726:,
722:Φ(
715:→
711:×
669:.
630:,
344:A
336:.
316:,
297:,
81:.
50:,
2852:.
2846:.
2816::
2782:.
2703:.
2665:X
2657:G
2636:G
2623:C
2612:G
2608:K
2599:K
2591:C
2584:X
2576:X
2572:G
2568:C
2564:X
2560:C
2556:G
2548:C
2544:G
2536:C
2532:G
2520:G
2501:X
2496:X
2493:S
2483:G
2479:g
2467:g
2463:G
2449:1
2426:,
2423:]
2420:]
2417:x
2414:[
2411:)
2406:2
2402:g
2398:(
2392:[
2389:)
2384:1
2380:g
2376:(
2370:=
2367:]
2364:x
2361:[
2358:)
2353:2
2349:g
2343:1
2339:g
2335:(
2311:x
2308:=
2305:]
2302:x
2299:[
2296:)
2293:1
2290:(
2274:X
2270:x
2266:G
2262:2
2259:g
2255:1
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2248:X
2244:X
2240:X
2236:G
2228:X
2221:G
2173:K
2146:V
2142:V
2128:V
2124:W
2094:3
2089:2
2085:x
2062:3
2057:1
2053:x
2032:)
2026:(
2003:V
1981:3
1977:S
1954:.
1949:3
1945:x
1941:,
1936:2
1932:x
1928:,
1923:1
1919:x
1898:V
1872:.
1866:2
1862:3
1855:=
1852:)
1849:u
1846:(
1838:=
1835:b
1831:,
1825:2
1822:1
1813:=
1810:)
1807:u
1804:(
1796:=
1793:a
1765:]
1759:a
1754:b
1744:b
1739:a
1733:[
1728:=
1724:)
1719:2
1715:u
1711:(
1701:]
1695:a
1690:b
1683:b
1675:a
1669:[
1664:=
1660:)
1657:u
1654:(
1644:]
1638:1
1633:0
1626:0
1621:1
1615:[
1610:=
1606:)
1603:1
1600:(
1571:2
1566:R
1554:3
1551:C
1534:.
1529:]
1523:1
1518:0
1511:0
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1500:u
1493:[
1488:=
1484:)
1479:2
1475:u
1471:(
1461:]
1455:1
1450:0
1443:0
1438:u
1432:[
1427:=
1423:)
1420:u
1417:(
1407:]
1401:1
1396:0
1389:0
1384:1
1378:[
1373:=
1369:)
1366:1
1363:(
1334:2
1329:C
1317:3
1314:C
1290:.
1285:]
1277:2
1273:u
1267:0
1260:0
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1249:[
1244:=
1240:)
1235:2
1231:u
1227:(
1217:]
1211:u
1206:0
1199:0
1194:1
1188:[
1183:=
1179:)
1176:u
1173:(
1163:]
1157:1
1152:0
1145:0
1140:1
1134:[
1129:=
1125:)
1122:1
1119:(
1090:2
1085:C
1069:u
1065:u
1061:3
1058:C
1054:u
1050:u
1026:.
1023:)
1020:g
1017:(
1011:=
1006:1
992:)
989:g
986:(
963:,
961:G
957:g
952:W
948:V
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928:)
926:W
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913:)
911:V
907:G
903:ρ
898:W
894:V
890:K
881:e
873:)
871:V
867:G
842:.
838:}
833:d
830:i
826:=
823:)
820:g
817:(
808:G
802:g
798:{
794:=
769:ρ
765:G
761:G
757:ρ
748:.
742:)
740:v
736:g
734:(
732:ρ
728:v
724:g
717:V
713:V
709:G
703:ρ
699:V
695:G
683:V
675:G
667:K
650:n
644:n
634:)
632:K
628:n
622:V
618:V
610:n
606:V
599:V
587:V
579:V
562:.
559:G
551:2
547:g
543:,
538:1
534:g
524:,
521:)
516:2
512:g
508:(
502:)
497:1
493:g
489:(
483:=
480:)
475:2
471:g
465:1
461:g
457:(
427:)
424:V
421:(
416:L
413:G
406:G
387:V
379:V
375:G
367:K
360:V
353:G
246:C
242:R
225:.
209:.
177:.
165:p
161:p
23:.
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