899:
705:
259:
867:
823:
778:
571:
535:
499:
331:
295:
455:
411:
384:
597:
357:
959:
940:
745:
725:
183:
163:
605:
933:
969:
964:
974:
926:
64:
44:
188:
33:
828:
783:
750:
540:
504:
460:
300:
264:
261:
are two finite-dimensional semisimple representations such that the characteristic polynomials of
72:
416:
389:
362:
501:, then the condition on the characteristic polynomials can be changed to the condition that Tr
76:
906:
576:
336:
29:
910:
730:
710:
168:
148:
25:
953:
700:{\displaystyle \rho :Gal(K^{\rm {sep}}/K)\to GL_{n}({\overline {\mathbb {Q} }}_{l})}
37:
51:
states that a character whose order is divisible by the highest power of a prime
898:
780:. Then the representation is uniquely determined by the values of the traces of
68:
17:
55:
dividing the order of a finite group remains irreducible when reduced mod
727:-adic representations of the absolute Galois group of some field
880:
Representation theory of finite groups and associative algebras
59:
and vanishes on all elements whose order is divisible by
914:
831:
786:
753:
733:
713:
608:
579:
543:
507:
463:
419:
392:
365:
339:
303:
267:
191:
171:
151:
24:can refer to several different theorems proved by
861:
817:
772:
739:
719:
699:
591:
565:
529:
493:
449:
405:
378:
351:
325:
289:
253:
177:
157:
747:, unramified outside some finite set of primes
49:Brauer–Nesbitt theorem on blocks of defect zero
869:(also using the Chebotarev density theorem).
254:{\displaystyle \rho _{i}:G\to GL_{n}(E),i=1,2}
934:
8:
106:-modules which are finite dimensional over
71:. A block of defect zero contains only one
941:
927:
847:
842:
830:
806:
785:
764:
752:
732:
712:
688:
679:
678:
676:
666:
645:
632:
631:
607:
578:
548:
542:
512:
506:
462:
418:
397:
391:
370:
364:
338:
308:
302:
272:
266:
218:
196:
190:
170:
150:
960:Representation theory of finite groups
889:Ann. of Math. (2) 42, (1941). 556-590.
7:
895:
893:
887:On the modular characters of groups.
413:are isomorphic representations. If
86:is a field of characteristic zero,
639:
636:
633:
14:
897:
862:{\displaystyle p\in M_{K}^{0}-S}
818:{\displaystyle \rho (Frob_{p})}
82:Another version states that if
812:
790:
773:{\displaystyle S\subset M_{K}}
694:
672:
656:
653:
624:
560:
554:
524:
518:
482:
476:
438:
432:
320:
314:
284:
278:
230:
224:
208:
1:
707:be a semisimple (continuous)
45:modular representation theory
913:. You can help Knowledge by
683:
566:{\displaystyle \rho _{2}(g)}
530:{\displaystyle \rho _{1}(g)}
494:{\displaystyle char(E)>n}
326:{\displaystyle \rho _{2}(g)}
290:{\displaystyle \rho _{1}(g)}
63:. Moreover, it belongs to a
991:
892:
450:{\displaystyle char(E)=0}
406:{\displaystyle \rho _{2}}
379:{\displaystyle \rho _{1}}
970:Theorems in group theory
885:Brauer, R.; Nesbitt, C.
965:Theorems about algebras
975:Abstract algebra stubs
909:-related article is a
863:
819:
774:
741:
721:
701:
602:As a consequence, let
593:
592:{\displaystyle g\in G}
567:
531:
495:
451:
407:
380:
353:
352:{\displaystyle g\in G}
327:
291:
255:
179:
159:
22:Brauer–Nesbitt theorem
864:
820:
775:
742:
722:
702:
594:
568:
532:
496:
452:
408:
381:
354:
328:
292:
256:
180:
160:
34:representation theory
829:
784:
751:
731:
711:
606:
577:
541:
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461:
417:
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363:
337:
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265:
189:
169:
149:
852:
859:
838:
815:
770:
737:
717:
697:
589:
563:
527:
491:
447:
403:
376:
349:
323:
287:
251:
185:be some field. If
175:
155:
138:are isomorphic as
122:as elements of Hom
73:ordinary character
922:
921:
740:{\displaystyle K}
720:{\displaystyle l}
686:
333:coincide for all
178:{\displaystyle E}
158:{\displaystyle G}
77:modular character
982:
943:
936:
929:
907:abstract algebra
901:
894:
878:Curtis, Reiner,
868:
866:
865:
860:
851:
846:
824:
822:
821:
816:
811:
810:
779:
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776:
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743:
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726:
724:
723:
718:
706:
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703:
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687:
682:
677:
671:
670:
649:
644:
643:
642:
598:
596:
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590:
572:
570:
569:
564:
553:
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536:
534:
533:
528:
517:
516:
500:
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497:
492:
456:
454:
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448:
412:
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401:
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223:
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184:
182:
181:
176:
164:
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156:
30:Cecil J. Nesbitt
990:
989:
985:
984:
983:
981:
980:
979:
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949:
948:
947:
875:
827:
826:
802:
782:
781:
760:
749:
748:
729:
728:
709:
708:
675:
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627:
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603:
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544:
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538:
508:
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502:
459:
458:
415:
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393:
388:
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366:
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360:
335:
334:
304:
299:
298:
268:
263:
262:
214:
192:
187:
186:
167:
166:
165:be a group and
147:
146:
125:
121:
115:
102:are semisimple
12:
11:
5:
988:
986:
978:
977:
972:
967:
962:
952:
951:
946:
945:
938:
931:
923:
920:
919:
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855:
850:
845:
841:
837:
834:
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789:
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736:
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685:
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669:
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658:
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648:
641:
638:
635:
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623:
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582:
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369:
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250:
247:
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229:
226:
221:
217:
213:
210:
207:
204:
199:
195:
174:
154:
123:
117:
111:
26:Richard Brauer
13:
10:
9:
6:
4:
3:
2:
987:
976:
973:
971:
968:
966:
963:
961:
958:
957:
955:
944:
939:
937:
932:
930:
925:
924:
918:
916:
912:
908:
903:
900:
896:
888:
884:
882:, Wiley 1962.
881:
877:
876:
872:
870:
856:
853:
848:
843:
839:
835:
832:
807:
803:
799:
796:
793:
787:
765:
761:
757:
754:
734:
714:
689:
667:
663:
659:
650:
646:
628:
621:
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615:
612:
609:
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586:
583:
580:
557:
549:
545:
521:
513:
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488:
485:
479:
473:
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467:
464:
444:
441:
435:
429:
426:
423:
420:
398:
394:
371:
367:
346:
343:
340:
317:
309:
305:
281:
273:
269:
248:
245:
242:
239:
236:
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211:
205:
202:
197:
193:
172:
152:
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141:
137:
133:
129:
120:
114:
109:
105:
101:
97:
93:
89:
85:
80:
78:
75:and only one
74:
70:
66:
62:
58:
54:
50:
46:
41:
39:
38:finite groups
35:
31:
27:
23:
19:
915:expanding it
904:
886:
879:
601:
144:
139:
135:
131:
127:
118:
112:
107:
103:
99:
95:
91:
87:
83:
81:
60:
56:
52:
48:
42:
21:
15:
69:defect zero
18:mathematics
954:Categories
873:References
142:-modules.
130:,k), then
94:-algebra,
854:−
836:∈
788:ρ
758:⊂
684:¯
657:→
610:ρ
584:∈
546:ρ
510:ρ
395:ρ
368:ρ
344:∈
306:ρ
270:ρ
209:→
194:ρ
573:for all
110:, and Tr
359:, then
32:in the
47:, the
20:, the
905:This
90:is a
65:block
911:stub
825:for
486:>
386:and
297:and
145:Let
134:and
116:= Tr
28:and
599:.
537:=Tr
457:or
67:of
43:In
36:of
16:In
956::
98:,
79:.
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942:e
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917:.
857:S
849:0
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840:M
833:p
813:)
808:p
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797:r
794:F
791:(
766:K
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755:S
735:K
715:l
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690:l
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673:(
668:n
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651:K
647:/
640:p
637:e
634:s
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587:G
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558:g
555:(
550:2
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519:(
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480:E
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439:)
436:E
433:(
430:r
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424:h
421:c
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318:g
315:(
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282:g
279:(
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243:1
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237:i
234:,
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140:A
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