25:
87:
856:
has an irreducible complex character of degree 2 which is not expressible as a non-negative integer combination of characters induced from linear characters of subgroups). An ingredient of the proof of Brauer's induction theorem is that when
238:
showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups.
712:
Brauer's induction theorem was proved in 1946, and there are now many alternative proofs. In 1986, Victor Snaith gave a proof by a radically different approach, topological in nature (an application of the
785:
362:. Note that in Artin's theorem the characters are induced from the trivial character of the cyclic group, while here they are induced from arbitrary characters (in applications to Artin's
298:
811:
475:
703:
664:
549:
325:
356:
840:
integer combination of characters induced from linear characters of subgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of
104:
151:
123:
989:
745:
is a virtual character. This result, together with the fact that a virtual character θ is an irreducible character if and only if θ(1)
718:
130:
555:|-th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal
137:
994:
119:
215:
203:
234:|, but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared,
717:). There has been related recent work on the question of finding natural and explicit forms of Brauer's theorem, notably by
817:) gives a means of constructing irreducible characters without explicitly constructing the associated representations.
752:
366:
functions it is important that the groups are cyclic and hence all characters are linear giving that the corresponding
242:
Another result between Artin's induction theorem and Brauer's induction theorem, also due to Brauer and also known as
144:
938:
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170:
68:
46:
39:
257:
603:) which differ (elementwise) from the trivial character by (integer multiples of) a sufficiently high power of
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is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of
714:
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576:
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638:
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852:(although solvability alone does not guarantee such expressions- for example, the solvable group
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741:
is a virtual character if and only if its restriction to each Brauer elementary subgroup of
825:
508:. These are direct products of cyclic groups and groups whose order is a power of a prime.
948:
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8:
923:
934:
884:
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944:
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399:
199:
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829:
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Brauer's induction theorem shows that the character ring can be generated (as an
391:
191:
551:-combinations of irreducible characters, where ω is a primitive complex |
983:
438:
575:). Several proofs of the theorem, beginning with a proof due to Brauer and
516:
The proof of Brauer's induction theorem exploits the ring structure of Char(
849:
434:
Its multiplication is given by the elementwise product of class functions.
383:
925:
Explicit Brauer
Induction: With Applications to Algebra and Number Theory
195:
16:
Fundamental result in the branch of mathematics known as character theory
820:
An initial motivation for Brauer's induction theorem was application to
426:). It is a ring by virtue of the fact that the product of characters of
832:. Highly significant for that application is whether each character of
579:, show that the trivial character is in the analogously defined ideal
903:
Corrected reprint of the 1976 original, published by
Academic Press.
567:), so the proof reduces to showing that the trivial character is in
86:
482:
504:
could be chosen from a very restricted collection, now called
733:, Brauer's induction theorem leads easily to his fundamental
929:. Cambridge Studies in Advanced Mathematics. Vol. 40.
815:
inner product on the ring of complex-valued class functions
737:, which asserts that a complex-valued class function of
595:
at a time, and constructing integer-valued elements of
869:
is induced from a linear character of some subgroup.
793:
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358:are induced from characters of cyclic subgroups of
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922:
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607:Once this is achieved for every prime divisor of |
543:
469:
350:
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292:
780:{\displaystyle \langle \theta ,\theta \rangle =1}
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250:is the fact that the regular representation of
214:A precursor to Brauer's induction theorem was
878:
800:
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768:
756:
293:{\displaystyle 1+\sum \lambda _{i}\rho _{i}}
611:|, some manipulations with congruences and
591:) by concentrating attention on one prime
500:In fact, Brauer showed that the subgroups
865:, every complex irreducible character of
682:
666:-valued class function lies in the ideal
643:
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171:Learn how and when to remove this message
69:Learn how and when to remove this message
824:. It shows that those are built up from
120:"Brauer's theorem on induced characters"
32:This article includes a list of general
204:representation theory of a finite group
990:Representation theory of finite groups
982:
920:
674:) if its values are all divisible (in
635:). An auxiliary result here is that a
398:consisting of integer combinations of
184:Brauer's theorem on induced characters
230:Brauer's theorem removes the factor |
194:, is a basic result in the branch of
109:adding citations to reliable sources
80:
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914:
627:), place the trivial character in
38:it lacks sufficient corresponding
14:
1006:
995:Theorems in representation theory
881:Character Theory of Finite Groups
806:{\displaystyle \langle ,\rangle }
615:, again exploiting the fact that
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470:{\displaystyle \lambda _{H}^{G}}
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735:characterization of characters
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715:Lefschetz fixed-point theorem
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698:{\displaystyle \mathbb {Z} }
659:{\displaystyle \mathbb {Z} }
544:{\displaystyle \mathbb {Z} }
373:
320:{\displaystyle \lambda _{i}}
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506:Brauer elementary subgroups
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1011:
931:Cambridge University Press
188:Brauer's induction theorem
973:, appendix to chapter XVI
844:are so expressible, then
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370:functions are analytic).
351:{\displaystyle \rho _{i}}
216:Artin's induction theorem
956:
430:is again a character of
971:Algebraic Number Theory
489:and λ ranges over
53:more precise citations.
921:Snaith, V. P. (1994).
879:Isaacs, I.M. (1994) .
807:
781:
699:
660:
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420:generalized characters
400:irreducible characters
352:
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826:Dirichlet L-functions
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782:
731:Frobenius reciprocity
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472:
424:difference characters
353:
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753:
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105:improve this article
466:
418:(alternatively, as
828:, or more general
803:
777:
695:
656:
613:algebraic integers
541:
467:
452:
443:induced characters
416:virtual characters
406:) is known as the
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329:positive rationals
317:
290:
254:can be written as
190:, and named after
830:Hecke L-functions
822:Artin L-functions
491:linear characters
186:, often known as
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439:abelian group
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389:
386:and let Char(
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122: –
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116:Find sources:
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94:This article
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83:
82:
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62:
52:
48:
42:
41:
35:
30:
21:
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970:
969:Serge Lang,
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924:
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861:is a finite
858:
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845:
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838:non-negative
837:
833:
819:
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738:
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728:
725:Applications
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481:ranges over
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445:of the form
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384:finite group
379:
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210:Background
131:newspapers
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883:. Dover.
801:⟩
795:⟨
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766:θ
760:θ
757:⟨
690:ω
651:ω
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536:ω
483:subgroups
454:λ
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340:ρ
309:λ
282:ρ
272:λ
268:∑
202:, within
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161:July 2024
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