867:-regular elements. The (usual character-ring) inner product of the Brauer character of a projective indecomposable with any other Brauer character can thus be defined: this is 0 if the second Brauer character is that of the socle of a non-isomorphic projective indecomposable, and 1 if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreducible character in the character of the lift of a projective indecomposable is equal to the number of occurrences of the Brauer character of the socle of the projective indecomposable when the restriction of the ordinary character to
1071:
The defect group of a block is unique up to conjugacy and has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and Brauer irreducible characters agree on elements of
771:
of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomposables have non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is
763:
In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules with characteristic dividing the group order are rarely projective. Indeed, if a simple module is projective, then it is the only simple module in its block,
502:
The Brauer character of a representation determines its composition factors but not, in general, its equivalence type. The irreducible Brauer characters are those afforded by the simple modules. These are integral (though not necessarily non-negative) combinations of the restrictions to elements of
887:
may be decomposed as non-negative integer combinations of the irreducible Brauer characters. The integers involved can be placed in a matrix, with the ordinary irreducible characters assigned rows and the irreducible Brauer characters assigned columns. This is referred to as the
1168:
The easiest block structure to analyse with non-trivial defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer,
746:
also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the
1102:
The order of the defect group of a block has many arithmetical characterizations related to representation theory. It is the largest invariant factor of the Cartan matrix of the block, and occurs with multiplicity one. Also, the power of
764:
which is then isomorphic to the endomorphism algebra of the underlying vector space, a full matrix algebra. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine.
675:
In modular representation theory, while
Maschke's theorem does not hold when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as
772:
the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module).
1153:
of the module. For example, the vertex of each indecomposable module in a block is contained (up to conjugacy) in the defect group of the block, and no proper subgroup of the defect group has that property.
284:
353:
146:, the process of averaging over the group needed to prove Maschke's theorem breaks down, and representations need not be completely reducible. Much of the discussion below implicitly assumes that the field
767:
For the group algebra of a finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the
883:
of the projective indecomposable modules may be calculated as follows: Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible
742:-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its
1018:
1054:
523:
In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the
1180:
Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a
358:
Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the
1228:
929:
Since a projective indecomposable module in a given block has all its composition factors in that same block, each block has its own Cartan matrix.
1336:
898:. It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively. The product of the transpose of
1137:
The defect group of a block also has several characterizations in the more module-theoretic approach to block theory, building on the work of
499:
the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation.
495:. Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to
1170:
511:
of each ordinary irreducible character is uniquely expressible as a non-negative integer combination of irreducible Brauer characters.
922:-th projective indecomposable module. The Cartan matrix is non-singular; in fact, its determinant is a power of the characteristic of
230:
302:
1115:
dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of
1122:
Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the
177:
does not divide the order of the group, the representation theory is similar to that in characteristic 0. He also investigated
88:
684:
has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra
1177:, among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block.
659:) of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime
688:
as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when
1312:
1290:
859:, and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic
974:
1237:
855:
When a projective module is lifted, the associated character vanishes on all elements of order divisible by
178:
1226:(1902), "On the Group Defined for any Given Field by the Multiplication Table of Any Given Finite Group",
1161:-subgroup as defect group is the same as the corresponding number for the normalizer in the group of that
1138:
507:
of the ordinary irreducible characters. Conversely, the restriction to the elements of order coprime to
433:
421:
154:
1196:. The indecomposable modules in wild blocks are extremely difficult to classify, even in principle.
775:
Each projective indecomposable module (and hence each projective module) in positive characteristic
1130:
is in the defect group of a given block, then each irreducible character in that block vanishes at
46:
1307:. North-Holland Mathematical Library. Vol. 25. Amsterdam-New York: North-Holland Publishing.
1215:, Actualités Scientifiques et Industrielles, vol. 195, Paris: Hermann et cie, pp. 1–15,
1157:
Brauer's first main theorem states that the number of blocks of a finite group that have a given
1108:
532:
181:
of some finite groups. The systematic study of modular representations, when the characteristic
1341:
1331:
1223:
984:
748:
600:
456:
from about 1940 onwards to study in greater depth the relationships between the characteristic
441:
217:
1023:
918:-th row are the multiplicities of the respective simple modules as composition factors of the
554:
111:
using the theory developed by Brauer, was particularly useful in the classification program.
31:
27:
1174:
890:
716:
209:
135:
123:
100:
99:
were too small in an appropriate sense. Also, a general result on embedding of elements of
87:
using modular representation theory played an important role in early progress towards the
16:
Studies linear representations of finite groups over a field K of positive characteristic p
362:. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.
8:
768:
96:
39:
464:, especially as the latter relates to the embedding of, and relationships between, its
428:. In that case, there are finite-dimensional modules for the group algebra that are not
95:
whose characterization was not amenable to purely group-theoretic methods because their
1275:
1259:
1185:
884:
880:
743:
359:
61:
1308:
1286:
1270:
1251:
696:, the endomorphism ring of the vector space underlying the associated simple module.
429:
108:
60:, modular representations arise naturally in other branches of mathematics, such as
1241:
1189:
425:
80:
1216:
1282:
1210:
644:
221:
213:
1206:
1181:
572:, and there is much interplay between the module theory of the three algebras.
453:
437:
84:
1325:
1255:
1193:
907:
539:
406:
73:
69:
65:
1192:, and their structure has been broadly determined in a series of papers by
1119:
dividing the degrees of the ordinary irreducible characters in that block.
469:
390:
198:
166:
92:
57:
53:
35:
871:-regular elements is expressed as a sum of irreducible Brauer characters.
1300:
1057:
401:, endowed with algebra multiplication by extending the multiplication of
23:
1263:
792:
704:
524:
379:
104:
216:. Over every field of characteristic other than 2, there is always a
850:
1246:
1134:. This is one of many consequences of Brauer's second main theorem.
1076:, and the simple module is projective. At the other extreme, when
460:
representation theory, ordinary character theory and structure of
130:|, then modular representations are completely reducible, as with
692:
is sufficiently large: each block is a full matrix algebra over
670:
635:
In ordinary representation theory, the number of simple modules
279:{\displaystyle {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}
348:{\displaystyle {\begin{bmatrix}1&1\\0&1\end{bmatrix}}.}
779:
may be lifted to a module in characteristic 0. Using the ring
564:
is closely related both to the structure of the group algebra
472:
to problems not directly phrased in terms of representations.
1085:
791:
may be decomposed as a sum of mutually orthogonal primitive
914:; this is a symmetric matrix such that the entries in its
1107:
dividing the index of the defect group of a block is the
874:
699:
To obtain the blocks, the identity element of the group
1212:Über die Darstellung von Gruppen in Galoisschen Feldern
727:. The block corresponding to the primitive idempotent
568:
and to the structure of the semisimple group algebra
311:
239:
157:
suffices), otherwise some statements need refinement.
1026:
987:
583:-module, and, by a process often known informally as
305:
233:
224:
with only 1 or −1 occurring on the diagonal, such as
483:is algebraically closed of positive characteristic
189:and was continued by him for the next few decades.
1274:
1149:to an indecomposable module, defined in terms of
1048:
1012:
851:Some orthogonality relations for Brauer characters
815:that occurs in this decomposition. The idempotent
432:. By contrast, in the characteristic 0 case every
347:
296:, there are many other possible matrices, such as
278:
134:(characteristic 0) representations, by virtue of
1269:
1229:Transactions of the American Mathematical Society
487:, there is a bijection between roots of unity in
1323:
185:divides the order of the group, was started by
165:The earliest work on representation theory over
607:-module arises by extension of scalars from an
491:and complex roots of unity of order coprime to
452:Modular representation theory was developed by
475:Brauer introduced the notion now known as the
1095:is a defect group for the principal block of
671:Blocks and the structure of the group algebra
365:
719:of the group algebra over the maximal order
1299:
1072:order prime to the relevant characteristic
630:
468:-subgroups. Such results can be applied in
1305:The representation theory of finite groups
1245:
220:such that the matrix can be written as a
208:is equivalent to the problem of finding
1277:Linear Representations of Finite Groups
1222:
170:
1337:Representation theory of finite groups
1324:
1205:
875:Decomposition matrix and Cartan matrix
611:-module. In general, however, not all
416:is divisible by the characteristic of
186:
89:classification of finite simple groups
819:lifts to a primitive idempotent, say
758:
397:-basis consisting of the elements of
703:is decomposed as a sum of primitive
514:
447:
150:is sufficiently large (for example,
56:. As well as having applications to
579:-module naturally gives rise to an
13:
651:. In the modular case, the number
615:-modules arise as reductions (mod
595:-module. On the other hand, since
14:
1353:
799:. Each projective indecomposable
553:of characteristic 0, such as the
932:
197:Finding a representation of the
961:). Formally, it is the largest
1091:-subgroup of the finite group
1043:
1037:
1007:
1001:
945:, Brauer associated a certain
103:2 in finite groups called the
1:
1238:American Mathematical Society
1199:
894:, and is frequently labelled
795:(not necessarily central) of
783:as above, with residue field
20:Modular representation theory
643:) is equal to the number of
623:-modules. Those that do are
420:, the group algebra is not
79:Within finite group theory,
7:
811:for a primitive idempotent
545:of positive characteristic
138:. In the other case, when |
10:
1358:
803:-module is isomorphic to
787:, the identity element of
738:. For each indecomposable
603:, each finite-dimensional
434:irreducible representation
366:Ring theory interpretation
192:
160:
1126:-part of a group element
1013:{\displaystyle DC_{G}(D)}
957:is the characteristic of
1049:{\displaystyle C_{G}(D)}
949:-subgroup, known as its
631:Number of simple modules
1236:(3), Providence, R.I.:
1224:Dickson, Leonard Eugene
1145:-subgroup known as the
1109:greatest common divisor
731:is the two-sided ideal
549:and field of fractions
533:discrete valuation ring
444:, hence is projective.
1050:
1014:
906:itself results in the
827:, and the left module
601:principal ideal domain
442:regular representation
349:
280:
114:If the characteristic
32:linear representations
1151:relative projectivity
1141:, which associates a
1051:
1015:
973:for which there is a
941:of the group algebra
424:, hence has non-zero
350:
281:
201:of two elements over
173:who showed that when
28:representation theory
26:, and is the part of
1024:
985:
975:Brauer correspondent
891:decomposition matrix
405:by linearity) is an
303:
231:
212:whose square is the
155:algebraically closed
122:does not divide the
1080:has characteristic
885:ordinary characters
881:composition factors
835:has reduction (mod
744:composition factors
560:. The structure of
374:and a finite group
81:character-theoretic
1186:semidihedral group
1046:
1010:
910:, usually denoted
759:Projective modules
667:-regular classes.
430:projective modules
412:When the order of
360:Jordan normal form
345:
336:
276:
267:
179:modular invariants
83:results proved by
62:algebraic geometry
1271:Jean-Pierre Serre
1188:or (generalized)
1173:, J.A. Green and
1111:of the powers of
981:for the subgroup
680:. When the field
645:conjugacy classes
503:order coprime to
448:Brauer characters
136:Maschke's theorem
109:George Glauberman
97:Sylow 2-subgroups
91:, especially for
1349:
1318:
1296:
1280:
1266:
1249:
1219:
1190:quaternion group
1055:
1053:
1052:
1047:
1036:
1035:
1019:
1017:
1016:
1011:
1000:
999:
839:) isomorphic to
751:is known as the
663:, the so-called
531:over a complete
477:Brauer character
426:Jacobson radical
354:
352:
351:
346:
341:
340:
285:
283:
282:
277:
272:
271:
142:| ≡ 0 mod
52:, necessarily a
1357:
1356:
1352:
1351:
1350:
1348:
1347:
1346:
1322:
1321:
1315:
1293:
1283:Springer-Verlag
1247:10.2307/1986379
1202:
1031:
1027:
1025:
1022:
1021:
995:
991:
986:
983:
982:
935:
877:
853:
761:
753:principal block
673:
633:
585:reduction (mod
521:
515:Reduction (mod
450:
368:
335:
334:
329:
323:
322:
317:
307:
306:
304:
301:
300:
295:
266:
265:
257:
251:
250:
245:
235:
234:
232:
229:
228:
222:diagonal matrix
214:identity matrix
207:
195:
163:
22:is a branch of
17:
12:
11:
5:
1355:
1345:
1344:
1339:
1334:
1320:
1319:
1313:
1297:
1291:
1267:
1220:
1201:
1198:
1182:dihedral group
1045:
1042:
1039:
1034:
1030:
1009:
1006:
1003:
998:
994:
990:
937:To each block
934:
931:
876:
873:
852:
849:
760:
757:
749:trivial module
672:
669:
632:
629:
558:-adic integers
520:
513:
454:Richard Brauer
449:
446:
438:direct summand
385:(which is the
370:Given a field
367:
364:
356:
355:
344:
339:
333:
330:
328:
325:
324:
321:
318:
316:
313:
312:
310:
293:
287:
286:
275:
270:
264:
261:
258:
256:
253:
252:
249:
246:
244:
241:
240:
238:
205:
194:
191:
171:Dickson (1902)
162:
159:
85:Richard Brauer
47:characteristic
15:
9:
6:
4:
3:
2:
1354:
1343:
1342:Finite fields
1340:
1338:
1335:
1333:
1332:Module theory
1330:
1329:
1327:
1316:
1314:0-444-86155-6
1310:
1306:
1302:
1298:
1294:
1292:0-387-90190-6
1288:
1284:
1279:
1278:
1272:
1268:
1265:
1261:
1257:
1253:
1248:
1243:
1239:
1235:
1231:
1230:
1225:
1221:
1218:
1214:
1213:
1208:
1204:
1203:
1197:
1195:
1194:Karin Erdmann
1191:
1187:
1183:
1178:
1176:
1175:J.G. Thompson
1172:
1166:
1164:
1160:
1155:
1152:
1148:
1144:
1140:
1135:
1133:
1129:
1125:
1120:
1118:
1114:
1110:
1106:
1100:
1098:
1094:
1090:
1087:
1083:
1079:
1075:
1069:
1067:
1063:
1059:
1040:
1032:
1028:
1004:
996:
992:
988:
980:
976:
972:
968:
964:
960:
956:
952:
948:
944:
940:
933:Defect groups
930:
927:
925:
921:
917:
913:
909:
908:Cartan matrix
905:
901:
897:
893:
892:
886:
882:
872:
870:
866:
862:
858:
848:
846:
842:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
773:
770:
765:
756:
754:
750:
745:
741:
737:
734:
730:
726:
722:
718:
714:
710:
706:
702:
697:
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691:
687:
683:
679:
668:
666:
662:
658:
654:
650:
646:
642:
638:
628:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
588:
582:
578:
573:
571:
567:
563:
559:
557:
552:
548:
544:
541:
540:residue field
537:
534:
530:
527:of the group
526:
525:group algebra
518:
512:
510:
506:
500:
498:
494:
490:
486:
482:
478:
473:
471:
467:
463:
459:
455:
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443:
439:
435:
431:
427:
423:
419:
415:
410:
408:
407:Artinian ring
404:
400:
396:
392:
388:
384:
381:
380:group algebra
377:
373:
363:
361:
342:
337:
331:
326:
319:
314:
308:
299:
298:
297:
292:
273:
268:
262:
259:
254:
247:
242:
236:
227:
226:
225:
223:
219:
215:
211:
204:
200:
190:
188:
187:Brauer (1935)
184:
180:
176:
172:
168:
167:finite fields
158:
156:
153:
149:
145:
141:
137:
133:
129:
125:
121:
117:
112:
110:
106:
102:
98:
94:
93:simple groups
90:
86:
82:
77:
75:
74:number theory
71:
70:combinatorics
67:
66:coding theory
63:
59:
55:
51:
48:
44:
41:
37:
36:finite groups
33:
30:that studies
29:
25:
21:
1304:
1276:
1233:
1227:
1211:
1179:
1167:
1162:
1158:
1156:
1150:
1146:
1142:
1136:
1131:
1127:
1123:
1121:
1116:
1112:
1104:
1101:
1096:
1092:
1088:
1081:
1077:
1073:
1070:
1065:
1061:
978:
970:
966:
962:
958:
954:
951:defect group
950:
946:
942:
938:
936:
928:
923:
919:
915:
911:
903:
899:
895:
889:
878:
868:
864:
860:
856:
854:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
800:
796:
788:
784:
780:
776:
774:
766:
762:
752:
739:
735:
732:
728:
724:
720:
712:
708:
700:
698:
693:
689:
685:
681:
677:
674:
664:
660:
656:
652:
648:
640:
636:
634:
624:
620:
616:
612:
608:
604:
596:
592:
586:
584:
580:
576:
574:
569:
565:
561:
555:
550:
546:
542:
535:
528:
522:
516:
508:
504:
501:
496:
492:
488:
484:
480:
476:
474:
470:group theory
465:
461:
457:
451:
417:
413:
411:
402:
398:
394:
391:vector space
386:
382:
375:
371:
369:
357:
290:
288:
202:
199:cyclic group
196:
182:
174:
164:
151:
147:
143:
139:
131:
127:
119:
115:
113:
107:, proved by
78:
58:group theory
54:prime number
49:
45:of positive
42:
19:
18:
1301:Walter Feit
1240:: 285–301,
1165:-subgroup.
1139:J. A. Green
1058:centralizer
793:idempotents
705:idempotents
24:mathematics
1326:Categories
1207:Brauer, R.
1200:References
965:-subgroup
863:module on
422:semisimple
105:Z* theorem
1256:0002-9947
1171:E.C. Dade
260:−
1303:(1982).
1273:(1977).
1209:(1935),
1020:, where
715:), the
625:liftable
210:matrices
132:ordinary
1264:1986379
1056:is the
953:(where
591:, to a
479:. When
440:of the
193:Example
161:History
38:over a
1311:
1289:
1262:
1254:
1217:review
1147:vertex
1084:, the
717:center
678:blocks
378:, the
169:is by
1260:JSTOR
1086:Sylow
902:with
823:, of
769:socle
619:) of
599:is a
575:Each
538:with
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