4088:
3762:
4083:{\displaystyle {\begin{aligned}\left\langle \theta ^{G},\psi ^{G}\right\rangle &=\left\langle \left(\theta ^{G}\right)_{K},\psi \right\rangle \\&=\sum _{t\in T}\left\langle \left(\left_{t^{-1}Ht\cap K}\right)^{K},\psi \right\rangle \\&=\sum _{t\in T}\left\langle \left(\theta ^{t}\right)_{t^{-1}Ht\cap K},\psi _{t^{-1}Ht\cap K}\right\rangle ,\end{aligned}}}
2262:
2070:
2538:, have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by
1912:
2095:
495:
5162:
3370:
3671:
1463:
1317:
1163:
3022:
2742:
954:
4689:
894:
1955:
4608:
4432:
299:
1009:
4885:
1634:
1569:. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by the identity. Therefore, the first column contains the degree of each irreducible character.
3767:
2257:{\displaystyle \sum _{\chi _{i}}\chi _{i}(g){\overline {\chi _{i}(h)}}={\begin{cases}\left|C_{G}(g)\right|,&{\mbox{ if }}g,h{\mbox{ are conjugate }}\\0&{\mbox{ otherwise.}}\end{cases}}}
1807:
3214:
2842:
751:
1397:
4815:
4759:
2643:
2597:
1490:
in a compact form. Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of
4216:
One can find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of
3534:
398:
2486:
5283:
5060:
5036:
5012:
4941:
4913:
4508:
4460:
5072:
3453:, but is a powerful tool in the character theory and representation theory of finite groups. Its basic form concerns the way a character (or module) induced from a subgroup
3233:
4968:
4786:
4535:
4346:
5236:
3550:
1541:
5256:
5189:
133:(a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of
1567:
5209:
4988:
4484:
4366:
4299:
1169:
1015:
2949:
2348:
is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). More generally, the sum of the squares of the
4729:
4709:
4319:
4275:
2499:, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of
3386:
This alternative description of the induced character sometimes allows explicit computation from relatively little information about the embedding of
2648:
3732:
Mackey decomposition, in conjunction with
Frobenius reciprocity, yields a well-known and useful formula for the inner product of two class functions
900:
5620:
5562:. Graduate Texts in Mathematics. Vol. 42. Translated from the second French edition by Leonard L. Scott. New York-Heidelberg: Springer-Verlag.
4617:
840:
118:
74:
5575:
5541:
5518:
5481:
5443:
2065:{\displaystyle \left\langle \chi _{i},\chi _{j}\right\rangle ={\begin{cases}0&{\mbox{ if }}i\neq j,\\1&{\mbox{ if }}i=j.\end{cases}}}
5063:
2311:
are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.
4543:
77:
entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a
101:
developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of
5500:
117:
encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the
1775:
The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes.
4374:
241:
5466:
960:
4823:
676:
3721:. There is a similar formula for the restriction of an induced module to a subgroup, which holds for representations over any
129:
involves intricate calculations with character values. Easier, but still essential, results that use character theory include
5415:
5169:
5427:
3434:
2496:
1907:{\displaystyle \left\langle \alpha ,\beta \right\rangle :={\frac {1}{|G|}}\sum _{g\in G}\alpha (g){\overline {\beta (g)}}}
1784:
1578:
1333:
572:
349:
138:
102:
30:
This article is about the use of the term character theory in mathematics. For related senses of the word character, see
146:
3158:
5419:
5294:
2352:
of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class.
337:
114:
4236:
2751:
1949:
for the space of class-functions, and this yields the orthogonality relation for the rows of the character table:
126:
2908:
4173:
2428:
is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that
778:
558:
170:
94:
1769:
490:{\displaystyle \ker \chi _{\rho }:=\left\lbrace g\in G\mid \chi _{\rho }(g)=\chi _{\rho }(1)\right\rbrace ,}
31:
3086:. The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions.
626:
is the degree (that is, the dimension of the associated vector space) of the representation with character
4916:
4791:
4734:
4248:
3102:
2602:
2556:
1500:
534:
309:
62:
54:
3491:
2921:
2865:
2364:
554:
5312:
in 1937, yields information about the restriction of a complex irreducible character of a finite group
5157:{\displaystyle \chi _{\rho }(H)=\sum _{\lambda }m_{\lambda }e^{\lambda (H)},\quad H\in {\mathfrak {h}}}
4117:
are linear characters, in which case all the inner products appearing in the right hand sum are either
3365:{\displaystyle \theta ^{G}(h)=\sum _{i\ :\ t_{i}ht_{i}^{-1}\in H}\theta \left(t_{i}ht_{i}^{-1}\right).}
2443:
130:
5264:
5041:
5017:
4993:
4922:
4894:
4489:
4441:
773:
635:
199:
69:. The character carries the essential information about the representation in a more condensed form.
66:
50:
4761:. If we have a Lie group representation and an associated Lie algebra representation, the character
2171:
2000:
807:. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of
4946:
4764:
4513:
4324:
4252:
4217:
3666:{\displaystyle \left(\theta ^{G}\right)_{K}=\sum _{t\in T}\left(\left_{t^{-1}Ht\cap K}\right)^{K},}
2845:
2410:
1357:
177:
90:
58:
5214:
3417:
The general technique of character induction and later refinements found numerous applications in
5299:
3722:
2744:
defines a new linear representation. This gives rise to a group of linear characters, called the
2321:
Constructing the complete character table when only some of the irreducible characters are known.
1458:{\displaystyle \rho \otimes \rho =\left(\rho \wedge \rho \right)\oplus {\textrm {Sym}}^{2}\rho .}
122:
1511:
5571:
5537:
5514:
5496:
5477:
5457:
5439:
5323:
5241:
5174:
2861:
2553:, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if
2511:
2324:
Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
1946:
1931:
754:
653:
576:
1312:{\displaystyle \chi _{{\scriptscriptstyle {\rm {{Sym}^{2}}}}\rho }(g)={\tfrac {1}{2}}\!\left}
1158:{\displaystyle \chi _{{\scriptscriptstyle {\rm {{Alt}^{2}}}}\rho }(g)={\tfrac {1}{2}}\!\left}
5563:
5431:
4228:
4188:
2849:
2515:
1546:
1391:
1369:
788:
579:, then the corresponding character is the sum of the characters of those subrepresentations.
5585:
5453:
5194:
4973:
4469:
4351:
4284:
3017:{\displaystyle \langle \theta ^{G},\chi \rangle _{G}=\langle \theta ,\chi _{H}\rangle _{H}}
5581:
5449:
5305:
2745:
1473:
800:
522:
389:
70:
5513:(Corrected reprint of the 1976 original, published by Academic Press. ed.). Dover.
5556:
5529:
5309:
4714:
4694:
4304:
4260:
4213:. Accordingly, one can view the other values of the character as "twisted" dimensions.
3426:
3141:
2550:
2522:
2492:
2349:
1790:
1479:
1347:
619:
565:
518:
150:
134:
98:
78:
5474:
Moonshine beyond the
Monster: The Bridge Connecting Algebra, Modular Forms and Physics
17:
5614:
4232:
2500:
2433:
1798:
5328:
5295:
Irreducible representation § Applications in theoretical physics and chemistry
3418:
2318:
Decomposing an unknown character as a linear combination of irreducible characters.
1640:
173:
142:
82:
42:
5285:
of the) character can be computed more explicitly by the Weyl character formula.
5551:
4463:
4221:
3430:
3383:, this value does not depend on the particular choice of coset representatives.
2737:{\displaystyle \rho _{1}\otimes \rho _{2}(g)=(\rho _{1}(g)\otimes \rho _{2}(g))}
2296:
2089:, applying the same inner product to the columns of the character table yields:
86:
38:
3097:, Frobenius later gave an explicit way to construct a matrix representation of
2870:
The characters discussed in this section are assumed to be complex-valued. Let
949:{\displaystyle \chi _{\rho \otimes \sigma }=\chi _{\rho }\cdot \chi _{\sigma }}
5604:
5567:
5435:
3422:
2539:
5461:
2929:
form an orthonormal basis for the space of complex-valued class functions of
4278:
3450:
3446:
2402:
is the intersection of the kernels of some of the irreducible characters of
1494:. The columns are labelled by (representatives of) the conjugacy classes of
4109:-double coset representatives, as before). This formula is often used when
3748:, whose utility lies in the fact that it only depends on how conjugates of
3394:, and is often useful for calculation of particular character tables. When
4684:{\displaystyle \chi _{\rho }(\operatorname {Ad} _{g}(X))=\chi _{\rho }(X)}
5495:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
5493:
Lie groups, Lie algebras, and representations: An elementary introduction
3726:
1945:. With respect to this inner product, the irreducible characters form an
889:{\displaystyle \chi _{\rho \oplus \sigma }=\chi _{\rho }+\chi _{\sigma }}
587:
3062:
5430:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
4172:
One may interpret the character of a representation as the "twisted"
3421:
and elsewhere in mathematics, in the hands of mathematicians such as
525:. More precisely, the set of irreducible characters of a given group
4176:. Treating the character as a function of the elements of the group
5600:
2436:
iff each conjugacy class is a singleton iff the character table of
4603:{\displaystyle \chi _{\rho }(X)=\operatorname {Tr} (e^{\rho (X)})}
4156:
are both trivial characters, then the inner product simplifies to
3145:
3114:. This led to an alternative description of the induced character
2508:
3132:, it is only now necessary to describe its values on elements of
2314:
The orthogonality relations can aid many computations including:
1498:. It is customary to label the first row by the character of the
3461:
behaves on restriction back to a (possibly different) subgroup
2417:
is the intersection of the kernels of the linear characters of
4788:
of the Lie algebra representation is related to the character
4427:{\displaystyle \chi _{\rho }(g)=\operatorname {Tr} (\rho (g))}
294:{\displaystyle \chi _{\rho }(g)=\operatorname {Tr} (\rho (g))}
89:) by its character. The situation with representations over a
1004:{\displaystyle \chi _{\rho ^{*}}={\overline {\chi _{\rho }}}}
97:, so-called "modular representations", is more delicate, but
4880:{\displaystyle \chi _{\rho }(X)=\mathrm {X} _{\rho }(e^{X})}
3756:
intersect each other. The formula (with its derivation) is:
2507:
The character table does not in general determine the group
5302:, a combinatorial generalization of group-character theory.
4235:, and replacing the dimension with the character gives the
2363:
is simple) can be recognised from its character table. The
2250:
2058:
3725:, and has applications in a wide variety of algebraic and
3039:
and the rightmost inner product is for class functions of
557:
representations have the same characters. Over a field of
4231:
of an infinite-dimensional graded representation of the
2267:
where the sum is over all of the irreducible characters
3494:
3035:(the leftmost inner product is for class functions of
2241:
2225:
2209:
2037:
2009:
1486:
which encodes much useful information about the group
1222:
1179:
1068:
1025:
521:, that is, they each take a constant value on a given
5267:
5244:
5217:
5197:
5177:
5075:
5044:
5020:
4996:
4976:
4949:
4925:
4897:
4826:
4794:
4767:
4737:
4717:
4697:
4620:
4546:
4516:
4492:
4472:
4444:
4377:
4354:
4327:
4307:
4287:
4263:
3765:
3553:
3445:
The Mackey decomposition was defined and explored by
3236:
3161:
3128:. Since the induced character is a class function of
3120:. This induced character vanishes on all elements of
2952:
2754:
2651:
2605:
2559:
2446:
2098:
1958:
1810:
1581:
1549:
1514:
1400:
1172:
1018:
963:
903:
843:
679:
401:
244:
4125:, depending on whether or not the linear characters
1629:{\displaystyle C_{3}=\langle u\mid u^{3}=1\rangle ,}
500:
which is precisely the kernel of the representation
356:; in characteristic zero this is equal to the value
5555:
5534:Representations and Characters of Groups (2nd ed.)
5277:
5250:
5230:
5203:
5183:
5156:
5054:
5030:
5006:
4982:
4962:
4935:
4907:
4879:
4809:
4780:
4753:
4723:
4703:
4683:
4602:
4529:
4502:
4478:
4454:
4426:
4360:
4340:
4313:
4293:
4269:
4082:
3665:
3528:
3364:
3208:
3016:
2836:
2736:
2637:
2591:
2480:
2256:
2064:
1906:
1628:
1561:
1535:
1457:
1311:
1157:
1003:
948:
888:
745:
489:
293:
3402:, the induced character obtained is known as the
2464:
2460:
1233:
1079:
3209:{\displaystyle G=Ht_{1}\cup \ldots \cup Ht_{n},}
795:, then the number of irreducible characters of
8:
3005:
2985:
2973:
2953:
2856:Induced characters and Frobenius reciprocity
2837:{\displaystyle (g)=\chi _{1}(g)\chi _{2}(g)}
1620:
1595:
746:{\displaystyle {\frac {\chi (x)}{\chi (1)}}}
4817:of the group representation by the formula
3043:). Since the restriction of a character of
3124:which are not conjugate to any element of
61:that associates to each group element the
5269:
5268:
5266:
5243:
5222:
5216:
5196:
5176:
5148:
5147:
5122:
5112:
5102:
5080:
5074:
5046:
5045:
5043:
5022:
5021:
5019:
4998:
4997:
4995:
4975:
4954:
4948:
4927:
4926:
4924:
4899:
4898:
4896:
4868:
4855:
4850:
4831:
4825:
4801:
4796:
4793:
4772:
4766:
4745:
4744:
4736:
4716:
4696:
4666:
4638:
4625:
4619:
4582:
4551:
4545:
4521:
4515:
4494:
4493:
4491:
4471:
4446:
4445:
4443:
4382:
4376:
4368:is defined precisely as for any group as
4353:
4332:
4326:
4306:
4286:
4262:
4243:Characters of Lie groups and Lie algebras
4045:
4040:
4010:
4005:
3995:
3969:
3938:
3911:
3906:
3896:
3865:
3834:
3824:
3792:
3779:
3766:
3764:
3654:
3627:
3622:
3612:
3586:
3573:
3563:
3552:
3505:
3493:
3345:
3340:
3327:
3298:
3293:
3280:
3263:
3241:
3235:
3197:
3175:
3160:
3065:combination of irreducible characters of
3008:
2998:
2976:
2960:
2951:
2819:
2800:
2775:
2762:
2753:
2716:
2694:
2669:
2656:
2650:
2629:
2610:
2604:
2583:
2564:
2558:
2488:iff each irreducible character is linear.
2473:
2465:
2455:
2447:
2445:
2340:can be deduced from its character table:
2240:
2224:
2208:
2183:
2166:
2142:
2135:
2120:
2108:
2103:
2097:
2036:
2008:
1995:
1981:
1968:
1957:
1883:
1859:
1847:
1839:
1833:
1809:
1608:
1586:
1580:
1548:
1513:
1443:
1437:
1436:
1399:
1295:
1282:
1269:
1249:
1221:
1194:
1183:
1181:
1180:
1178:
1177:
1171:
1141:
1128:
1115:
1095:
1067:
1040:
1029:
1027:
1026:
1024:
1023:
1017:
990:
984:
973:
968:
962:
940:
927:
908:
902:
880:
867:
848:
842:
711:
693:
678:
593:, then the result is also a character of
464:
442:
412:
400:
249:
243:
5384:
5382:
3469:, and makes use of the decomposition of
1649:
5558:Linear Representations of Finite Groups
5356:
5062:can easily be computed in terms of the
4486:a finite-dimensional representation of
4301:a finite-dimensional representation of
4239:for each element of the Monster group.
2911:showed how to construct a character of
5389:
5367:, Springer-Verlag, 2012, Chap. 8, p392
5343:) is a real number for all characters
5038:. The restriction of the character to
3055:, this definition makes it clear that
2925:. Since the irreducible characters of
834:. Then the following identities hold:
119:classification of finite simple groups
75:representation theory of finite groups
5424:Representation theory. A first course
4191:is the dimension of the space, since
564:, two representations are isomorphic
541:-vector space of all class functions
7:
5401:
4810:{\displaystyle \mathrm {X} _{\rho }}
4754:{\displaystyle X\in {\mathfrak {g}}}
3544:, then Mackey's formula states that
2638:{\displaystyle \rho _{2}:G\to V_{2}}
2592:{\displaystyle \rho _{1}:G\to V_{1}}
1482:characters of a finite group form a
363:. A character of degree 1 is called
27:Concept in mathematical group theory
5270:
5149:
5047:
5023:
4999:
4928:
4900:
4746:
4495:
4447:
3529:{\textstyle G=\bigcup _{t\in T}HtK}
2933:, there is a unique class function
1508:on a 1-dimensional vector space by
652:, every such character value is an
582:If a character of the finite group
4851:
4797:
3740:induced from respective subgroups
2874:be a subgroup of the finite group
2491:It follows, using some results of
1643:with three elements and generator
1190:
1187:
1184:
1036:
1033:
1030:
830:Let ρ and σ be representations of
25:
5511:Character Theory of Finite Groups
4970:of an irreducible representation
2645:are linear representations, then
2549:are themselves a group under the
2481:{\displaystyle |G|\!\times \!|G|}
1504:, which is the trivial action of
508:a group homomorphism in general.
3437:, as well as Frobenius himself.
2336:Certain properties of the group
5621:Representation theory of groups
5278:{\displaystyle {\mathfrak {h}}}
5140:
5055:{\displaystyle {\mathfrak {h}}}
5031:{\displaystyle {\mathfrak {h}}}
5014:is determined by its values on
5007:{\displaystyle {\mathfrak {g}}}
4936:{\displaystyle {\mathfrak {h}}}
4908:{\displaystyle {\mathfrak {g}}}
4503:{\displaystyle {\mathfrak {g}}}
4455:{\displaystyle {\mathfrak {g}}}
3540:is a complex class function of
3027:for each irreducible character
2394:; this is a normal subgroup of
2327:Finding the order of the group.
1572:Here is the character table of
5536:. Cambridge University Press.
5132:
5126:
5092:
5086:
4874:
4861:
4843:
4837:
4678:
4672:
4656:
4653:
4647:
4631:
4597:
4592:
4586:
4575:
4563:
4557:
4510:, we can define the character
4421:
4418:
4412:
4406:
4394:
4388:
3253:
3247:
3101:, known as the representation
3089:Given a matrix representation
3069:, so is indeed a character of
2831:
2825:
2812:
2806:
2790:
2784:
2781:
2755:
2731:
2728:
2722:
2706:
2700:
2687:
2681:
2675:
2622:
2576:
2545:The linear representations of
2474:
2466:
2456:
2448:
2195:
2189:
2154:
2148:
2132:
2126:
1895:
1889:
1880:
1874:
1848:
1840:
1524:
1518:
1301:
1288:
1261:
1255:
1215:
1209:
1147:
1134:
1107:
1101:
1061:
1055:
737:
731:
723:
717:
708:
705:
699:
680:
476:
470:
454:
448:
288:
285:
279:
273:
261:
255:
1:
5428:Graduate Texts in Mathematics
4963:{\displaystyle \chi _{\rho }}
4943:. The value of the character
4781:{\displaystyle \chi _{\rho }}
4530:{\displaystyle \chi _{\rho }}
4341:{\displaystyle \chi _{\rho }}
4135:have the same restriction to
3108:, and written analogously as
2919:, using what is now known as
2844:. This group is connected to
2497:modular representation theory
1785:Schur orthogonality relations
568:they have the same character.
375:has characteristic zero, the
5231:{\displaystyle m_{\lambda }}
4711:in the associated Lie group
3398:is the trivial character of
2158:
1899:
1789:The space of complex-valued
996:
504:. However, the character is
147:generalized quaternion group
4614:The character will satisfy
4174:dimension of a vector space
571:If a representation is the
115:irreducible representations
5637:
5168:where the sum is over all
4246:
2895:denote its restriction to
2859:
2398:. Each normal subgroup of
2332:Character table properties
1782:
1536:{\displaystyle \rho (g)=1}
1471:
811:(and they even divide if
799:is equal to the number of
338:irreducible representation
29:
5568:10.1007/978-1-4684-9458-7
5436:10.1007/978-1-4612-0979-9
3682:is the class function of
3536:is a disjoint union, and
2909:Ferdinand Georg Frobenius
2359:(and thus whether or not
2295:denotes the order of the
2227: are conjugate
1394:, which is determined by
5251:{\displaystyle \lambda }
5184:{\displaystyle \lambda }
3051:is again a character of
2355:All normal subgroups of
5491:Hall, Brian C. (2015),
5238:is the multiplicity of
4919:with Cartan subalgebra
3379:is a class function of
3219:then, given an element
2943:with the property that
2371:is the set of elements
1779:Orthogonality relations
121:. Close to half of the
103:modular representations
41:, more specifically in
32:Character (mathematics)
5472:Gannon, Terry (2006).
5279:
5252:
5232:
5205:
5185:
5158:
5056:
5032:
5008:
4984:
4964:
4937:
4917:semisimple Lie algebra
4909:
4881:
4811:
4782:
4755:
4725:
4705:
4685:
4604:
4531:
4504:
4480:
4456:
4428:
4362:
4342:
4315:
4295:
4271:
4249:Weyl character formula
4084:
3667:
3530:
3366:
3210:
3018:
2838:
2738:
2639:
2593:
2482:
2258:
2066:
1908:
1630:
1563:
1562:{\displaystyle g\in G}
1537:
1501:trivial representation
1459:
1313:
1159:
1005:
950:
890:
747:
673:is irreducible, then
642:. In particular, when
600:Every character value
491:
295:
141:stating that a finite
18:Orthogonality relation
5509:Isaacs, I.M. (1994).
5316:to a normal subgroup
5280:
5253:
5233:
5206:
5204:{\displaystyle \rho }
5186:
5159:
5057:
5033:
5009:
4985:
4983:{\displaystyle \rho }
4965:
4938:
4910:
4882:
4812:
4783:
4756:
4726:
4706:
4686:
4605:
4532:
4505:
4481:
4479:{\displaystyle \rho }
4457:
4429:
4363:
4361:{\displaystyle \rho }
4343:
4316:
4296:
4294:{\displaystyle \rho }
4272:
4237:McKay–Thompson series
4085:
3668:
3531:
3404:permutation character
3367:
3211:
3019:
2922:Frobenius reciprocity
2866:Frobenius reciprocity
2839:
2739:
2640:
2594:
2483:
2259:
2067:
1909:
1772:third root of unity.
1631:
1564:
1538:
1472:Further information:
1460:
1314:
1160:
1006:
951:
891:
826:Arithmetic properties
748:
492:
296:
127:Feit–Thompson theorem
85:is determined (up to
65:of the corresponding
5265:
5261:The (restriction to
5242:
5215:
5195:
5175:
5073:
5042:
5018:
4994:
4974:
4947:
4923:
4895:
4824:
4792:
4765:
4735:
4715:
4695:
4618:
4544:
4514:
4490:
4470:
4442:
4375:
4352:
4325:
4305:
4285:
4261:
3763:
3551:
3492:
3441:Mackey decomposition
3234:
3159:
2950:
2878:. Given a character
2846:Dirichlet characters
2752:
2748:under the operation
2649:
2603:
2557:
2444:
2096:
1956:
1808:
1579:
1547:
1512:
1398:
1170:
1016:
961:
901:
841:
787:does not divide the
774:algebraically closed
677:
399:
242:
81:representation of a
73:initially developed
51:group representation
5300:Association schemes
4253:Algebraic character
4218:monstrous moonshine
4187:, its value at the
4168:"Twisted" dimension
3419:finite group theory
3353:
3306:
2514:: for example, the
2411:commutator subgroup
1370:alternating product
1358:conjugate transpose
586:is restricted to a
5552:Serre, Jean-Pierre
5363:Nicolas Bourbaki,
5331:, a group element
5275:
5248:
5228:
5201:
5181:
5154:
5107:
5052:
5028:
5004:
4980:
4960:
4933:
4905:
4877:
4807:
4778:
4751:
4721:
4701:
4681:
4600:
4527:
4500:
4476:
4452:
4424:
4358:
4338:
4311:
4291:
4267:
4080:
4078:
3980:
3876:
3663:
3597:
3526:
3516:
3457:of a finite group
3449:in the context of
3410:(on the cosets of
3362:
3336:
3314:
3289:
3206:
3061:is a non-negative
3014:
2903:be a character of
2834:
2734:
2635:
2589:
2478:
2303:. Note that since
2254:
2249:
2245:
2229:
2213:
2115:
2062:
2057:
2041:
2013:
1904:
1870:
1793:of a finite group
1626:
1559:
1533:
1455:
1309:
1231:
1202:
1155:
1077:
1048:
1001:
946:
886:
743:
577:subrepresentations
487:
291:
171:finite-dimensional
131:Burnside's theorem
5577:978-0-387-90190-9
5543:978-0-521-00392-6
5520:978-0-486-68014-9
5483:978-0-521-83531-2
5445:978-0-387-97495-8
5404:Proposition 10.12
5324:Frobenius formula
5098:
4891:Suppose now that
4724:{\displaystyle G}
4704:{\displaystyle g}
4314:{\displaystyle G}
4270:{\displaystyle G}
4097:is a full set of
3965:
3861:
3582:
3501:
3275:
3269:
3259:
3073:. It is known as
2862:Induced character
2244:
2228:
2212:
2161:
2099:
2040:
2012:
1947:orthonormal basis
1932:complex conjugate
1902:
1855:
1853:
1762:
1761:
1440:
1230:
1076:
999:
801:conjugacy classes
755:algebraic integer
741:
654:algebraic integer
379:of the character
344:of the character
16:(Redirected from
5628:
5589:
5561:
5547:
5524:
5505:
5487:
5465:
5405:
5399:
5393:
5386:
5377:
5374:
5368:
5361:
5319:
5315:
5308:, introduced by
5284:
5282:
5281:
5276:
5274:
5273:
5257:
5255:
5254:
5249:
5237:
5235:
5234:
5229:
5227:
5226:
5210:
5208:
5207:
5202:
5190:
5188:
5187:
5182:
5163:
5161:
5160:
5155:
5153:
5152:
5136:
5135:
5117:
5116:
5106:
5085:
5084:
5061:
5059:
5058:
5053:
5051:
5050:
5037:
5035:
5034:
5029:
5027:
5026:
5013:
5011:
5010:
5005:
5003:
5002:
4989:
4987:
4986:
4981:
4969:
4967:
4966:
4961:
4959:
4958:
4942:
4940:
4939:
4934:
4932:
4931:
4914:
4912:
4911:
4906:
4904:
4903:
4886:
4884:
4883:
4878:
4873:
4872:
4860:
4859:
4854:
4836:
4835:
4816:
4814:
4813:
4808:
4806:
4805:
4800:
4787:
4785:
4784:
4779:
4777:
4776:
4760:
4758:
4757:
4752:
4750:
4749:
4730:
4728:
4727:
4722:
4710:
4708:
4707:
4702:
4690:
4688:
4687:
4682:
4671:
4670:
4643:
4642:
4630:
4629:
4609:
4607:
4606:
4601:
4596:
4595:
4556:
4555:
4536:
4534:
4533:
4528:
4526:
4525:
4509:
4507:
4506:
4501:
4499:
4498:
4485:
4483:
4482:
4477:
4461:
4459:
4458:
4453:
4451:
4450:
4433:
4431:
4430:
4425:
4387:
4386:
4367:
4365:
4364:
4359:
4347:
4345:
4344:
4339:
4337:
4336:
4321:, the character
4320:
4318:
4317:
4312:
4300:
4298:
4297:
4292:
4276:
4274:
4273:
4268:
4229:graded dimension
4224:
4212:
4186:
4163:
4155:
4151:
4147:
4134:
4130:
4124:
4120:
4116:
4112:
4108:
4096:
4089:
4087:
4086:
4081:
4079:
4072:
4068:
4067:
4066:
4053:
4052:
4032:
4031:
4018:
4017:
4004:
4000:
3999:
3979:
3958:
3954:
3950:
3943:
3942:
3937:
3933:
3932:
3919:
3918:
3905:
3901:
3900:
3875:
3854:
3850:
3846:
3839:
3838:
3833:
3829:
3828:
3802:
3798:
3797:
3796:
3784:
3783:
3755:
3751:
3747:
3743:
3739:
3735:
3720:
3716:
3712:
3690:
3681:
3672:
3670:
3669:
3664:
3659:
3658:
3653:
3649:
3648:
3635:
3634:
3621:
3617:
3616:
3596:
3578:
3577:
3572:
3568:
3567:
3543:
3539:
3535:
3533:
3532:
3527:
3515:
3485:-double cosets.
3484:
3472:
3468:
3464:
3460:
3456:
3413:
3409:
3401:
3397:
3393:
3389:
3382:
3378:
3371:
3369:
3368:
3363:
3358:
3354:
3352:
3344:
3332:
3331:
3313:
3305:
3297:
3285:
3284:
3273:
3267:
3246:
3245:
3226:
3222:
3215:
3213:
3212:
3207:
3202:
3201:
3180:
3179:
3151:
3139:
3136:. If one writes
3135:
3131:
3127:
3123:
3119:
3113:
3107:
3100:
3096:
3092:
3085:
3079:
3075:the character of
3072:
3068:
3060:
3054:
3050:
3047:to the subgroup
3046:
3042:
3038:
3034:
3030:
3023:
3021:
3020:
3015:
3013:
3012:
3003:
3002:
2981:
2980:
2965:
2964:
2942:
2938:
2932:
2928:
2918:
2914:
2906:
2902:
2898:
2894:
2885:
2881:
2877:
2873:
2850:Fourier analysis
2843:
2841:
2840:
2835:
2824:
2823:
2805:
2804:
2780:
2779:
2767:
2766:
2743:
2741:
2740:
2735:
2721:
2720:
2699:
2698:
2674:
2673:
2661:
2660:
2644:
2642:
2641:
2636:
2634:
2633:
2615:
2614:
2598:
2596:
2595:
2590:
2588:
2587:
2569:
2568:
2548:
2537:
2528:
2520:
2516:quaternion group
2487:
2485:
2484:
2479:
2477:
2469:
2459:
2451:
2439:
2431:
2427:
2420:
2416:
2405:
2401:
2397:
2393:
2378:
2374:
2370:
2362:
2358:
2347:
2339:
2310:
2306:
2302:
2294:
2279:
2275:
2263:
2261:
2260:
2255:
2253:
2252:
2246:
2243: otherwise.
2242:
2230:
2226:
2214:
2210:
2202:
2198:
2188:
2187:
2162:
2157:
2147:
2146:
2136:
2125:
2124:
2114:
2113:
2112:
2088:
2084:
2071:
2069:
2068:
2063:
2061:
2060:
2042:
2038:
2014:
2010:
1991:
1987:
1986:
1985:
1973:
1972:
1944:
1929:
1928:
1913:
1911:
1910:
1905:
1903:
1898:
1884:
1869:
1854:
1852:
1851:
1843:
1834:
1829:
1825:
1796:
1767:
1758:
1753:
1746:
1741:
1729:
1722:
1717:
1712:
1700:
1695:
1690:
1685:
1676:
1667:
1658:
1650:
1635:
1633:
1632:
1627:
1613:
1612:
1591:
1590:
1568:
1566:
1565:
1560:
1542:
1540:
1539:
1534:
1507:
1497:
1493:
1489:
1478:The irreducible
1468:Character tables
1464:
1462:
1461:
1456:
1448:
1447:
1442:
1441:
1438:
1431:
1427:
1392:symmetric square
1389:
1385:
1367:
1363:
1355:
1345:
1331:
1318:
1316:
1315:
1310:
1308:
1304:
1300:
1299:
1287:
1286:
1274:
1273:
1268:
1264:
1254:
1253:
1232:
1223:
1208:
1207:
1203:
1201:
1200:
1199:
1198:
1193:
1164:
1162:
1161:
1156:
1154:
1150:
1146:
1145:
1133:
1132:
1120:
1119:
1114:
1110:
1100:
1099:
1078:
1069:
1054:
1053:
1049:
1047:
1046:
1045:
1044:
1039:
1010:
1008:
1007:
1002:
1000:
995:
994:
985:
980:
979:
978:
977:
955:
953:
952:
947:
945:
944:
932:
931:
919:
918:
895:
893:
892:
887:
885:
884:
872:
871:
859:
858:
833:
820:
810:
806:
798:
794:
786:
771:
764:
760:
752:
750:
749:
744:
742:
740:
726:
712:
698:
697:
672:
668:
651:
641:
633:
629:
625:
617:
614:
610:
596:
592:
585:
563:
550:
540:
532:
528:
503:
496:
494:
493:
488:
483:
479:
469:
468:
447:
446:
417:
416:
387:
374:
370:
362:
355:
347:
335:
323:
307:
300:
298:
297:
292:
254:
253:
234:
218:is the function
217:
209:
205:
197:
182:
168:
154:
21:
5636:
5635:
5631:
5630:
5629:
5627:
5626:
5625:
5611:
5610:
5597:
5592:
5578:
5550:
5544:
5530:Liebeck, Martin
5528:James, Gordon;
5527:
5521:
5508:
5503:
5490:
5484:
5471:
5446:
5416:Fulton, William
5414:
5409:
5408:
5400:
5396:
5387:
5380:
5375:
5371:
5362:
5358:
5353:
5317:
5313:
5306:Clifford theory
5291:
5263:
5262:
5240:
5239:
5218:
5213:
5212:
5193:
5192:
5173:
5172:
5118:
5108:
5076:
5071:
5070:
5040:
5039:
5016:
5015:
4992:
4991:
4972:
4971:
4950:
4945:
4944:
4921:
4920:
4893:
4892:
4864:
4849:
4827:
4822:
4821:
4795:
4790:
4789:
4768:
4763:
4762:
4733:
4732:
4713:
4712:
4693:
4692:
4662:
4634:
4621:
4616:
4615:
4578:
4547:
4542:
4541:
4517:
4512:
4511:
4488:
4487:
4468:
4467:
4440:
4439:
4378:
4373:
4372:
4350:
4349:
4328:
4323:
4322:
4303:
4302:
4283:
4282:
4259:
4258:
4255:
4245:
4222:
4205:
4192:
4177:
4170:
4157:
4153:
4149:
4136:
4132:
4126:
4122:
4118:
4114:
4110:
4098:
4094:
4077:
4076:
4041:
4036:
4006:
3991:
3987:
3986:
3985:
3981:
3956:
3955:
3907:
3892:
3888:
3887:
3883:
3882:
3881:
3877:
3852:
3851:
3820:
3816:
3815:
3814:
3810:
3803:
3788:
3775:
3774:
3770:
3761:
3760:
3753:
3749:
3745:
3741:
3737:
3733:
3718:
3714:
3692:
3683:
3677:
3623:
3608:
3604:
3603:
3599:
3598:
3559:
3555:
3554:
3549:
3548:
3541:
3537:
3490:
3489:
3474:
3470:
3466:
3462:
3458:
3454:
3443:
3411:
3407:
3399:
3395:
3391:
3387:
3380:
3376:
3323:
3322:
3318:
3276:
3237:
3232:
3231:
3224:
3220:
3193:
3171:
3157:
3156:
3149:
3137:
3133:
3129:
3125:
3121:
3115:
3109:
3105:
3098:
3094:
3090:
3083:
3077:
3070:
3066:
3056:
3052:
3048:
3044:
3040:
3036:
3032:
3028:
3004:
2994:
2972:
2956:
2948:
2947:
2940:
2934:
2930:
2926:
2916:
2912:
2904:
2900:
2896:
2892:
2887:
2883:
2879:
2875:
2871:
2868:
2860:Main articles:
2858:
2815:
2796:
2771:
2758:
2750:
2749:
2746:character group
2712:
2690:
2665:
2652:
2647:
2646:
2625:
2606:
2601:
2600:
2579:
2560:
2555:
2554:
2546:
2536:
2530:
2526:
2518:
2442:
2441:
2437:
2429:
2425:
2418:
2414:
2403:
2399:
2395:
2380:
2376:
2372:
2368:
2367:of a character
2360:
2356:
2350:absolute values
2345:
2337:
2334:
2308:
2304:
2300:
2287:
2281:
2280:and the symbol
2277:
2273:
2268:
2248:
2247:
2238:
2232:
2231:
2206:
2179:
2178:
2174:
2167:
2138:
2137:
2116:
2104:
2094:
2093:
2086:
2076:
2056:
2055:
2034:
2028:
2027:
2006:
1996:
1977:
1964:
1963:
1959:
1954:
1953:
1935:
1919:
1918:
1885:
1838:
1815:
1811:
1806:
1805:
1794:
1791:class functions
1787:
1781:
1765:
1756:
1749:
1744:
1740:
1734:
1725:
1720:
1715:
1711:
1705:
1698:
1693:
1688:
1681:
1670:
1661:
1656:
1604:
1582:
1577:
1576:
1545:
1544:
1510:
1509:
1505:
1495:
1491:
1487:
1484:character table
1476:
1474:Character table
1470:
1435:
1417:
1413:
1396:
1395:
1387:
1372:
1365:
1361:
1351:
1337:
1323:
1291:
1278:
1245:
1244:
1240:
1239:
1238:
1234:
1182:
1173:
1168:
1167:
1137:
1124:
1091:
1090:
1086:
1085:
1084:
1080:
1028:
1019:
1014:
1013:
986:
969:
964:
959:
958:
936:
923:
904:
899:
898:
876:
863:
844:
839:
838:
831:
828:
812:
808:
804:
796:
792:
777:
769:
762:
758:
727:
713:
689:
675:
674:
670:
660:
643:
639:
631:
627:
623:
615:
612:
601:
594:
590:
583:
561:
542:
538:
530:
526:
523:conjugacy class
519:class functions
517:Characters are
514:
501:
460:
438:
425:
421:
408:
397:
396:
390:normal subgroup
385:
380:
372:
368:
357:
353:
345:
333:
321:
316:
305:
245:
240:
239:
224:
219:
215:
207:
203:
184:
180:
166:
163:
152:
111:
71:Georg Frobenius
35:
28:
23:
22:
15:
12:
11:
5:
5634:
5632:
5624:
5623:
5613:
5612:
5609:
5608:
5596:
5595:External links
5593:
5591:
5590:
5576:
5548:
5542:
5525:
5519:
5506:
5502:978-3319134666
5501:
5488:
5482:
5469:
5444:
5410:
5407:
5406:
5394:
5378:
5369:
5355:
5354:
5352:
5349:
5348:
5347:
5326:
5321:
5310:A. H. Clifford
5303:
5297:
5290:
5287:
5272:
5247:
5225:
5221:
5200:
5180:
5166:
5165:
5151:
5146:
5143:
5139:
5134:
5131:
5128:
5125:
5121:
5115:
5111:
5105:
5101:
5097:
5094:
5091:
5088:
5083:
5079:
5066:, as follows:
5049:
5025:
5001:
4979:
4957:
4953:
4930:
4902:
4889:
4888:
4876:
4871:
4867:
4863:
4858:
4853:
4848:
4845:
4842:
4839:
4834:
4830:
4804:
4799:
4775:
4771:
4748:
4743:
4740:
4720:
4700:
4680:
4677:
4674:
4669:
4665:
4661:
4658:
4655:
4652:
4649:
4646:
4641:
4637:
4633:
4628:
4624:
4612:
4611:
4599:
4594:
4591:
4588:
4585:
4581:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4554:
4550:
4524:
4520:
4497:
4475:
4449:
4438:Meanwhile, if
4436:
4435:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4385:
4381:
4357:
4335:
4331:
4310:
4290:
4266:
4244:
4241:
4203:
4169:
4166:
4091:
4090:
4075:
4071:
4065:
4062:
4059:
4056:
4051:
4048:
4044:
4039:
4035:
4030:
4027:
4024:
4021:
4016:
4013:
4009:
4003:
3998:
3994:
3990:
3984:
3978:
3975:
3972:
3968:
3964:
3961:
3959:
3957:
3953:
3949:
3946:
3941:
3936:
3931:
3928:
3925:
3922:
3917:
3914:
3910:
3904:
3899:
3895:
3891:
3886:
3880:
3874:
3871:
3868:
3864:
3860:
3857:
3855:
3853:
3849:
3845:
3842:
3837:
3832:
3827:
3823:
3819:
3813:
3809:
3806:
3804:
3801:
3795:
3791:
3787:
3782:
3778:
3773:
3769:
3768:
3674:
3673:
3662:
3657:
3652:
3647:
3644:
3641:
3638:
3633:
3630:
3626:
3620:
3615:
3611:
3607:
3602:
3595:
3592:
3589:
3585:
3581:
3576:
3571:
3566:
3562:
3558:
3525:
3522:
3519:
3514:
3511:
3508:
3504:
3500:
3497:
3442:
3439:
3427:Richard Brauer
3373:
3372:
3361:
3357:
3351:
3348:
3343:
3339:
3335:
3330:
3326:
3321:
3317:
3312:
3309:
3304:
3301:
3296:
3292:
3288:
3283:
3279:
3272:
3266:
3262:
3258:
3255:
3252:
3249:
3244:
3240:
3217:
3216:
3205:
3200:
3196:
3192:
3189:
3186:
3183:
3178:
3174:
3170:
3167:
3164:
3142:disjoint union
3025:
3024:
3011:
3007:
3001:
2997:
2993:
2990:
2987:
2984:
2979:
2975:
2971:
2968:
2963:
2959:
2955:
2890:
2857:
2854:
2833:
2830:
2827:
2822:
2818:
2814:
2811:
2808:
2803:
2799:
2795:
2792:
2789:
2786:
2783:
2778:
2774:
2770:
2765:
2761:
2757:
2733:
2730:
2727:
2724:
2719:
2715:
2711:
2708:
2705:
2702:
2697:
2693:
2689:
2686:
2683:
2680:
2677:
2672:
2668:
2664:
2659:
2655:
2632:
2628:
2624:
2621:
2618:
2613:
2609:
2586:
2582:
2578:
2575:
2572:
2567:
2563:
2551:tensor product
2534:
2523:dihedral group
2505:
2504:
2493:Richard Brauer
2489:
2476:
2472:
2468:
2463:
2458:
2454:
2450:
2422:
2407:
2353:
2333:
2330:
2329:
2328:
2325:
2322:
2319:
2285:
2271:
2265:
2264:
2251:
2239:
2237:
2234:
2233:
2223:
2220:
2217:
2211: if
2207:
2205:
2201:
2197:
2194:
2191:
2186:
2182:
2177:
2173:
2172:
2170:
2165:
2160:
2156:
2153:
2150:
2145:
2141:
2134:
2131:
2128:
2123:
2119:
2111:
2107:
2102:
2073:
2072:
2059:
2054:
2051:
2048:
2045:
2039: if
2035:
2033:
2030:
2029:
2026:
2023:
2020:
2017:
2011: if
2007:
2005:
2002:
2001:
1999:
1994:
1990:
1984:
1980:
1976:
1971:
1967:
1962:
1915:
1914:
1901:
1897:
1894:
1891:
1888:
1882:
1879:
1876:
1873:
1868:
1865:
1862:
1858:
1850:
1846:
1842:
1837:
1832:
1828:
1824:
1821:
1818:
1814:
1797:has a natural
1783:Main article:
1780:
1777:
1760:
1759:
1754:
1747:
1742:
1738:
1731:
1730:
1723:
1718:
1713:
1709:
1702:
1701:
1696:
1691:
1686:
1678:
1677:
1668:
1659:
1654:
1637:
1636:
1625:
1622:
1619:
1616:
1611:
1607:
1603:
1600:
1597:
1594:
1589:
1585:
1558:
1555:
1552:
1532:
1529:
1526:
1523:
1520:
1517:
1469:
1466:
1454:
1451:
1446:
1434:
1430:
1426:
1423:
1420:
1416:
1412:
1409:
1406:
1403:
1348:tensor product
1320:
1319:
1307:
1303:
1298:
1294:
1290:
1285:
1281:
1277:
1272:
1267:
1263:
1260:
1257:
1252:
1248:
1243:
1237:
1229:
1226:
1220:
1217:
1214:
1211:
1206:
1197:
1192:
1189:
1186:
1176:
1165:
1153:
1149:
1144:
1140:
1136:
1131:
1127:
1123:
1118:
1113:
1109:
1106:
1103:
1098:
1094:
1089:
1083:
1075:
1072:
1066:
1063:
1060:
1057:
1052:
1043:
1038:
1035:
1032:
1022:
1011:
998:
993:
989:
983:
976:
972:
967:
956:
943:
939:
935:
930:
926:
922:
917:
914:
911:
907:
896:
883:
879:
875:
870:
866:
862:
857:
854:
851:
847:
827:
824:
823:
822:
766:
739:
736:
733:
730:
725:
722:
719:
716:
710:
707:
704:
701:
696:
692:
688:
685:
682:
657:
620:roots of unity
598:
580:
569:
566:if and only if
559:characteristic
552:
513:
510:
498:
497:
486:
482:
478:
475:
472:
467:
463:
459:
456:
453:
450:
445:
441:
437:
434:
431:
428:
424:
420:
415:
411:
407:
404:
383:
371:is finite and
319:
302:
301:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
252:
248:
222:
200:representation
162:
159:
145:cannot have a
135:Richard Brauer
113:Characters of
110:
107:
99:Richard Brauer
95:characteristic
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5633:
5622:
5619:
5618:
5616:
5606:
5602:
5599:
5598:
5594:
5587:
5583:
5579:
5573:
5569:
5565:
5560:
5559:
5553:
5549:
5545:
5539:
5535:
5531:
5526:
5522:
5516:
5512:
5507:
5504:
5498:
5494:
5489:
5485:
5479:
5475:
5470:
5468:
5463:
5459:
5455:
5451:
5447:
5441:
5437:
5433:
5429:
5425:
5421:
5417:
5413:Lecture 2 of
5412:
5411:
5403:
5398:
5395:
5391:
5385:
5383:
5379:
5373:
5370:
5366:
5360:
5357:
5350:
5346:
5342:
5338:
5334:
5330:
5327:
5325:
5322:
5311:
5307:
5304:
5301:
5298:
5296:
5293:
5292:
5288:
5286:
5259:
5245:
5223:
5219:
5198:
5178:
5171:
5144:
5141:
5137:
5129:
5123:
5119:
5113:
5109:
5103:
5099:
5095:
5089:
5081:
5077:
5069:
5068:
5067:
5065:
5064:weight spaces
4977:
4955:
4951:
4918:
4915:is a complex
4869:
4865:
4856:
4846:
4840:
4832:
4828:
4820:
4819:
4818:
4802:
4773:
4769:
4741:
4738:
4718:
4698:
4675:
4667:
4663:
4659:
4650:
4644:
4639:
4635:
4626:
4622:
4589:
4583:
4579:
4572:
4569:
4566:
4560:
4552:
4548:
4540:
4539:
4538:
4522:
4518:
4473:
4465:
4415:
4409:
4403:
4400:
4397:
4391:
4383:
4379:
4371:
4370:
4369:
4355:
4333:
4329:
4308:
4288:
4280:
4264:
4254:
4250:
4242:
4240:
4238:
4234:
4233:Monster group
4230:
4226:
4219:
4214:
4210:
4206:
4199:
4195:
4190:
4184:
4180:
4175:
4167:
4165:
4161:
4146:
4142:
4139:
4129:
4106:
4102:
4073:
4069:
4063:
4060:
4057:
4054:
4049:
4046:
4042:
4037:
4033:
4028:
4025:
4022:
4019:
4014:
4011:
4007:
4001:
3996:
3992:
3988:
3982:
3976:
3973:
3970:
3966:
3962:
3960:
3951:
3947:
3944:
3939:
3934:
3929:
3926:
3923:
3920:
3915:
3912:
3908:
3902:
3897:
3893:
3889:
3884:
3878:
3872:
3869:
3866:
3862:
3858:
3856:
3847:
3843:
3840:
3835:
3830:
3825:
3821:
3817:
3811:
3807:
3805:
3799:
3793:
3789:
3785:
3780:
3776:
3771:
3759:
3758:
3757:
3730:
3728:
3724:
3710:
3706:
3702:
3699:
3695:
3689:
3686:
3680:
3660:
3655:
3650:
3645:
3642:
3639:
3636:
3631:
3628:
3624:
3618:
3613:
3609:
3605:
3600:
3593:
3590:
3587:
3583:
3579:
3574:
3569:
3564:
3560:
3556:
3547:
3546:
3545:
3523:
3520:
3517:
3512:
3509:
3506:
3502:
3498:
3495:
3486:
3482:
3478:
3452:
3448:
3447:George Mackey
3440:
3438:
3436:
3435:Michio Suzuki
3432:
3428:
3424:
3420:
3415:
3405:
3384:
3359:
3355:
3349:
3346:
3341:
3337:
3333:
3328:
3324:
3319:
3315:
3310:
3307:
3302:
3299:
3294:
3290:
3286:
3281:
3277:
3270:
3264:
3260:
3256:
3250:
3242:
3238:
3230:
3229:
3228:
3203:
3198:
3194:
3190:
3187:
3184:
3181:
3176:
3172:
3168:
3165:
3162:
3155:
3154:
3153:
3147:
3143:
3118:
3112:
3104:
3087:
3082:
3076:
3064:
3059:
3009:
2999:
2995:
2991:
2988:
2982:
2977:
2969:
2966:
2961:
2957:
2946:
2945:
2944:
2937:
2924:
2923:
2910:
2893:
2867:
2863:
2855:
2853:
2851:
2847:
2828:
2820:
2816:
2809:
2801:
2797:
2793:
2787:
2776:
2772:
2768:
2763:
2759:
2747:
2725:
2717:
2713:
2709:
2703:
2695:
2691:
2684:
2678:
2670:
2666:
2662:
2657:
2653:
2630:
2626:
2619:
2616:
2611:
2607:
2584:
2580:
2573:
2570:
2565:
2561:
2552:
2543:
2541:
2533:
2524:
2517:
2513:
2510:
2502:
2501:Graham Higman
2498:
2494:
2490:
2470:
2461:
2452:
2435:
2423:
2412:
2408:
2391:
2387:
2383:
2366:
2354:
2351:
2344:The order of
2343:
2342:
2341:
2331:
2326:
2323:
2320:
2317:
2316:
2315:
2312:
2298:
2292:
2288:
2274:
2235:
2221:
2218:
2215:
2203:
2199:
2192:
2184:
2180:
2175:
2168:
2163:
2151:
2143:
2139:
2129:
2121:
2117:
2109:
2105:
2100:
2092:
2091:
2090:
2083:
2079:
2052:
2049:
2046:
2043:
2031:
2024:
2021:
2018:
2015:
2003:
1997:
1992:
1988:
1982:
1978:
1974:
1969:
1965:
1960:
1952:
1951:
1950:
1948:
1942:
1938:
1933:
1926:
1922:
1892:
1886:
1877:
1871:
1866:
1863:
1860:
1856:
1844:
1835:
1830:
1826:
1822:
1819:
1816:
1812:
1804:
1803:
1802:
1800:
1799:inner product
1792:
1786:
1778:
1776:
1773:
1771:
1755:
1752:
1748:
1743:
1737:
1733:
1732:
1728:
1724:
1719:
1714:
1708:
1704:
1703:
1697:
1692:
1687:
1684:
1680:
1679:
1674:
1669:
1665:
1660:
1655:
1652:
1651:
1648:
1646:
1642:
1623:
1617:
1614:
1609:
1605:
1601:
1598:
1592:
1587:
1583:
1575:
1574:
1573:
1570:
1556:
1553:
1550:
1530:
1527:
1521:
1515:
1503:
1502:
1485:
1481:
1475:
1467:
1465:
1452:
1449:
1444:
1432:
1428:
1424:
1421:
1418:
1414:
1410:
1407:
1404:
1401:
1393:
1384:
1380:
1376:
1371:
1359:
1354:
1349:
1344:
1340:
1335:
1330:
1326:
1305:
1296:
1292:
1283:
1279:
1275:
1270:
1265:
1258:
1250:
1246:
1241:
1235:
1227:
1224:
1218:
1212:
1204:
1195:
1174:
1166:
1151:
1142:
1138:
1129:
1125:
1121:
1116:
1111:
1104:
1096:
1092:
1087:
1081:
1073:
1070:
1064:
1058:
1050:
1041:
1020:
1012:
991:
987:
981:
974:
970:
965:
957:
941:
937:
933:
928:
924:
920:
915:
912:
909:
905:
897:
881:
877:
873:
868:
864:
860:
855:
852:
849:
845:
837:
836:
835:
825:
819:
815:
802:
790:
784:
780:
775:
767:
756:
734:
728:
720:
714:
702:
694:
690:
686:
683:
667:
663:
658:
655:
650:
646:
637:
621:
608:
604:
599:
589:
581:
578:
574:
570:
567:
560:
556:
553:
549:
545:
536:
529:into a field
524:
520:
516:
515:
511:
509:
507:
484:
480:
473:
465:
461:
457:
451:
443:
439:
435:
432:
429:
426:
422:
418:
413:
409:
405:
402:
395:
394:
393:
391:
386:
378:
366:
360:
351:
343:
339:
331:
327:
322:
313:
311:
282:
276:
270:
267:
264:
258:
250:
246:
238:
237:
236:
233:
229:
225:
213:
201:
195:
191:
187:
179:
175:
172:
160:
158:
156:
148:
144:
140:
139:Michio Suzuki
136:
132:
128:
124:
120:
116:
108:
106:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
5557:
5533:
5510:
5492:
5473:
5423:
5397:
5372:
5364:
5359:
5344:
5340:
5336:
5332:
5329:Real element
5260:
5167:
4890:
4613:
4437:
4256:
4215:
4208:
4201:
4197:
4193:
4182:
4178:
4171:
4159:
4144:
4140:
4137:
4127:
4104:
4100:
4092:
3731:
3708:
3704:
3700:
3697:
3693:
3687:
3684:
3678:
3675:
3487:
3480:
3476:
3444:
3416:
3403:
3385:
3374:
3218:
3116:
3110:
3103:induced from
3088:
3081:induced from
3080:
3074:
3057:
3026:
2935:
2920:
2888:
2869:
2544:
2531:
2506:
2389:
2385:
2381:
2335:
2313:
2290:
2283:
2269:
2266:
2081:
2077:
2074:
1940:
1936:
1924:
1920:
1916:
1788:
1774:
1763:
1750:
1735:
1726:
1706:
1682:
1672:
1663:
1644:
1641:cyclic group
1638:
1571:
1499:
1483:
1477:
1382:
1378:
1374:
1356:denotes the
1352:
1342:
1338:
1328:
1324:
1321:
829:
817:
813:
782:
665:
661:
648:
644:
611:is a sum of
606:
602:
547:
543:
505:
499:
381:
376:
364:
358:
341:
329:
325:
317:
315:A character
314:
303:
231:
227:
220:
211:
193:
189:
185:
174:vector space
164:
143:simple group
112:
109:Applications
93:of positive
83:finite group
46:
43:group theory
36:
5420:Harris, Joe
5390:Gannon 2006
5376:Serre, §2.5
4464:Lie algebra
3727:topological
3691:defined by
3431:Walter Feit
3227:, we have:
2512:isomorphism
2297:centralizer
326:irreducible
202:of a group
161:Definitions
87:isomorphism
39:mathematics
5605:PlanetMath
5351:References
5335:such that
5211:and where
4247:See also:
4225:-invariant
4200:(1)) = Tr(
3729:contexts.
3451:Lie groups
3423:Emil Artin
2540:E. C. Dade
2529:elements,
2379:for which
1334:direct sum
573:direct sum
555:Isomorphic
512:Properties
324:is called
5601:Character
5462:246650103
5402:Hall 2015
5246:λ
5224:λ
5199:ρ
5179:λ
5145:∈
5124:λ
5114:λ
5104:λ
5100:∑
5082:ρ
5078:χ
4978:ρ
4956:ρ
4952:χ
4857:ρ
4833:ρ
4829:χ
4803:ρ
4774:ρ
4770:χ
4742:∈
4668:ρ
4664:χ
4645:
4627:ρ
4623:χ
4584:ρ
4573:
4553:ρ
4549:χ
4523:ρ
4519:χ
4474:ρ
4410:ρ
4404:
4384:ρ
4380:χ
4356:ρ
4334:ρ
4330:χ
4289:ρ
4279:Lie group
4196:(1) = Tr(
4061:∩
4047:−
4038:ψ
4026:∩
4012:−
3993:θ
3974:∈
3967:∑
3948:ψ
3927:∩
3913:−
3894:θ
3870:∈
3863:∑
3844:ψ
3822:θ
3790:ψ
3777:θ
3643:∩
3629:−
3610:θ
3591:∈
3584:∑
3561:θ
3510:∈
3503:⋃
3347:−
3316:θ
3308:∈
3300:−
3261:∑
3239:θ
3188:∪
3185:…
3182:∪
3144:of right
3006:⟩
2996:χ
2989:θ
2986:⟨
2974:⟩
2970:χ
2958:θ
2954:⟨
2817:χ
2798:χ
2773:χ
2769:∗
2760:χ
2714:ρ
2710:⊗
2692:ρ
2667:ρ
2663:⊗
2654:ρ
2623:→
2608:ρ
2577:→
2562:ρ
2462:×
2159:¯
2140:χ
2118:χ
2106:χ
2101:∑
2019:≠
1979:χ
1966:χ
1900:¯
1887:β
1872:α
1864:∈
1857:∑
1823:β
1817:α
1770:primitive
1621:⟩
1602:∣
1596:⟨
1554:∈
1516:ρ
1450:ρ
1433:⊕
1425:ρ
1422:∧
1419:ρ
1408:ρ
1405:⊗
1402:ρ
1284:ρ
1280:χ
1251:ρ
1247:χ
1205:ρ
1175:χ
1130:ρ
1126:χ
1122:−
1097:ρ
1093:χ
1051:ρ
1021:χ
997:¯
992:ρ
988:χ
975:∗
971:ρ
966:χ
942:σ
938:χ
934:⋅
929:ρ
925:χ
916:σ
913:⊗
910:ρ
906:χ
882:σ
878:χ
869:ρ
865:χ
856:σ
853:⊕
850:ρ
846:χ
729:χ
715:χ
466:ρ
462:χ
444:ρ
440:χ
436:∣
430:∈
414:ρ
410:χ
406:
350:dimension
277:ρ
271:
251:ρ
247:χ
235:given by
212:character
155:-subgroup
47:character
5615:Category
5554:(1977).
5532:(2001).
5422:(1991).
5289:See also
4731:and all
4691:for all
4207:) = dim(
4189:identity
4070:⟩
3983:⟨
3952:⟩
3879:⟨
3848:⟩
3812:⟨
3800:⟩
3772:⟨
3713:for all
3375:Because
2521:and the
1989:⟩
1961:⟨
1827:⟩
1813:⟨
1543:for all
757:for all
622:, where
588:subgroup
226: :
188: :
183:and let
55:function
5586:0450380
5454:1153249
5365:Algèbre
5170:weights
4227:is the
4093:(where
3063:integer
2434:abelian
2293:)|
1930:is the
1653:
1480:complex
1390:is the
1368:is the
1346:is the
1332:is the
634:is the
537:of the
533:form a
388:is the
367:. When
348:is the
340:. The
308:is the
176:over a
149:as its
125:of the
79:complex
57:on the
5584:
5574:
5540:
5517:
5499:
5480:
5467:online
5460:
5452:
5442:
4220:: the
4162:|
4158:|
3676:where
3274:
3268:
3152:, say
3146:cosets
2899:. Let
2886:, let
2365:kernel
2282:|
1917:where
1764:where
1364:, and
1322:where
753:is an
377:kernel
365:linear
342:degree
336:is an
330:simple
304:where
210:. The
151:Sylow
67:matrix
45:, the
4462:is a
4277:is a
4148:. If
3473:into
3140:as a
2915:from
2509:up to
2495:from
1768:is a
789:order
636:order
535:basis
310:trace
198:be a
192:→ GL(
178:field
169:be a
123:proof
91:field
63:trace
59:group
53:is a
49:of a
5572:ISBN
5538:ISBN
5515:ISBN
5497:ISBN
5478:ISBN
5458:OCLC
5440:ISBN
4466:and
4281:and
4251:and
4152:and
4131:and
4113:and
3752:and
3744:and
3736:and
3723:ring
3703:) =
3433:and
2864:and
2848:and
2599:and
2409:The
2388:) =
2307:and
2075:For
1639:the
1386:and
779:char
776:and
669:and
630:and
618:-th
165:Let
137:and
5603:at
5564:doi
5432:doi
5191:of
4990:of
4537:by
4348:of
4257:If
4121:or
3717:in
3488:If
3465:of
3414:).
3406:of
3390:in
3223:of
3148:of
3093:of
3031:of
2939:of
2882:of
2525:of
2440:is
2432:is
2424:If
2413:of
2392:(1)
2375:in
2299:of
2276:of
2085:in
1934:of
1657:(1)
1439:Sym
1388:Sym
1373:Alt
1366:Alt
1360:of
803:of
791:of
772:is
768:If
761:in
659:If
638:of
575:of
506:not
403:ker
361:(1)
352:of
332:if
328:or
214:of
206:on
37:In
5617::
5582:MR
5580:.
5570:.
5476:.
5456:.
5450:MR
5448:.
5438:.
5426:.
5418:;
5381:^
5258:.
4636:Ad
4570:Tr
4401:Tr
4164:.
4143:∩
4141:Ht
4103:,
3701:ht
3688:Ht
3479:,
3429:,
3425:,
2907:.
2852:.
2542:.
2503:).
2080:,
1831::=
1801::
1647::
1381:∧
1377:=
1350:,
1336:,
821:).
816:=
664:=
647:=
546:→
419::=
392::
312:.
306:Tr
268:Tr
230:→
157:.
105:.
5607:.
5588:.
5566::
5546:.
5523:.
5486:.
5464:.
5434::
5392:)
5388:(
5345:χ
5341:g
5339:(
5337:χ
5333:g
5320:.
5318:N
5314:G
5271:h
5220:m
5164:,
5150:h
5142:H
5138:,
5133:)
5130:H
5127:(
5120:e
5110:m
5096:=
5093:)
5090:H
5087:(
5048:h
5024:h
5000:g
4929:h
4901:g
4887:.
4875:)
4870:X
4866:e
4862:(
4852:X
4847:=
4844:)
4841:X
4838:(
4798:X
4747:g
4739:X
4719:G
4699:g
4679:)
4676:X
4673:(
4660:=
4657:)
4654:)
4651:X
4648:(
4640:g
4632:(
4610:.
4598:)
4593:)
4590:X
4587:(
4580:e
4576:(
4567:=
4564:)
4561:X
4558:(
4496:g
4448:g
4434:.
4422:)
4419:)
4416:g
4413:(
4407:(
4398:=
4395:)
4392:g
4389:(
4309:G
4265:G
4223:j
4211:)
4209:V
4204:V
4202:I
4198:ρ
4194:χ
4185:)
4183:g
4181:(
4179:χ
4160:T
4154:ψ
4150:θ
4145:K
4138:t
4133:ψ
4128:θ
4123:0
4119:1
4115:ψ
4111:θ
4107:)
4105:K
4101:H
4099:(
4095:T
4074:,
4064:K
4058:t
4055:H
4050:1
4043:t
4034:,
4029:K
4023:t
4020:H
4015:1
4008:t
4002:)
3997:t
3989:(
3977:T
3971:t
3963:=
3945:,
3940:K
3935:)
3930:K
3924:t
3921:H
3916:1
3909:t
3903:]
3898:t
3890:[
3885:(
3873:T
3867:t
3859:=
3841:,
3836:K
3831:)
3826:G
3818:(
3808:=
3794:G
3786:,
3781:G
3754:K
3750:H
3746:K
3742:H
3738:ψ
3734:θ
3719:H
3715:h
3711:)
3709:h
3707:(
3705:θ
3698:t
3696:(
3694:θ
3685:t
3679:θ
3661:,
3656:K
3651:)
3646:K
3640:t
3637:H
3632:1
3625:t
3619:]
3614:t
3606:[
3601:(
3594:T
3588:t
3580:=
3575:K
3570:)
3565:G
3557:(
3542:H
3538:θ
3524:K
3521:t
3518:H
3513:T
3507:t
3499:=
3496:G
3483:)
3481:K
3477:H
3475:(
3471:G
3467:G
3463:K
3459:G
3455:H
3412:H
3408:G
3400:H
3396:θ
3392:G
3388:H
3381:H
3377:θ
3360:.
3356:)
3350:1
3342:i
3338:t
3334:h
3329:i
3325:t
3320:(
3311:H
3303:1
3295:i
3291:t
3287:h
3282:i
3278:t
3271::
3265:i
3257:=
3254:)
3251:h
3248:(
3243:G
3225:H
3221:h
3204:,
3199:n
3195:t
3191:H
3177:1
3173:t
3169:H
3166:=
3163:G
3150:H
3138:G
3134:H
3130:G
3126:H
3122:G
3117:θ
3111:ρ
3106:ρ
3099:G
3095:H
3091:ρ
3084:θ
3078:G
3071:G
3067:G
3058:θ
3053:H
3049:H
3045:G
3041:H
3037:G
3033:G
3029:χ
3010:H
3000:H
2992:,
2983:=
2978:G
2967:,
2962:G
2941:G
2936:θ
2931:G
2927:G
2917:θ
2913:G
2905:H
2901:θ
2897:H
2891:H
2889:χ
2884:G
2880:χ
2876:G
2872:H
2832:)
2829:g
2826:(
2821:2
2813:)
2810:g
2807:(
2802:1
2794:=
2791:)
2788:g
2785:(
2782:]
2777:2
2764:1
2756:[
2732:)
2729:)
2726:g
2723:(
2718:2
2707:)
2704:g
2701:(
2696:1
2688:(
2685:=
2682:)
2679:g
2676:(
2671:2
2658:1
2631:2
2627:V
2620:G
2617::
2612:2
2585:1
2581:V
2574:G
2571::
2566:1
2547:G
2535:4
2532:D
2527:8
2519:Q
2475:|
2471:G
2467:|
2457:|
2453:G
2449:|
2438:G
2430:G
2426:G
2421:.
2419:G
2415:G
2406:.
2404:G
2400:G
2396:G
2390:χ
2386:g
2384:(
2382:χ
2377:G
2373:g
2369:χ
2361:G
2357:G
2346:G
2338:G
2309:h
2305:g
2301:g
2291:g
2289:(
2286:G
2284:C
2278:G
2272:i
2270:χ
2236:0
2222:h
2219:,
2216:g
2204:,
2200:|
2196:)
2193:g
2190:(
2185:G
2181:C
2176:|
2169:{
2164:=
2155:)
2152:h
2149:(
2144:i
2133:)
2130:g
2127:(
2122:i
2110:i
2087:G
2082:h
2078:g
2053:.
2050:j
2047:=
2044:i
2032:1
2025:,
2022:j
2016:i
2004:0
1998:{
1993:=
1983:j
1975:,
1970:i
1943:)
1941:g
1939:(
1937:β
1927:)
1925:g
1923:(
1921:β
1896:)
1893:g
1890:(
1881:)
1878:g
1875:(
1867:G
1861:g
1849:|
1845:G
1841:|
1836:1
1820:,
1795:G
1766:ω
1757:ω
1751:ω
1745:1
1739:2
1736:χ
1727:ω
1721:ω
1716:1
1710:1
1707:χ
1699:1
1694:1
1689:1
1683:1
1675:)
1673:u
1671:(
1666:)
1664:u
1662:(
1645:u
1624:,
1618:1
1615:=
1610:3
1606:u
1599:u
1593:=
1588:3
1584:C
1557:G
1551:g
1531:1
1528:=
1525:)
1522:g
1519:(
1506:G
1496:G
1492:G
1488:G
1453:.
1445:2
1429:)
1415:(
1411:=
1383:ρ
1379:ρ
1375:ρ
1362:ρ
1353:ρ
1343:σ
1341:⊗
1339:ρ
1329:σ
1327:⊕
1325:ρ
1306:]
1302:)
1297:2
1293:g
1289:(
1276:+
1271:2
1266:)
1262:)
1259:g
1256:(
1242:(
1236:[
1228:2
1225:1
1219:=
1216:)
1213:g
1210:(
1196:2
1191:m
1188:y
1185:S
1152:]
1148:)
1143:2
1139:g
1135:(
1117:2
1112:)
1108:)
1105:g
1102:(
1088:(
1082:[
1074:2
1071:1
1065:=
1062:)
1059:g
1056:(
1042:2
1037:t
1034:l
1031:A
982:=
921:=
874:+
861:=
832:G
818:C
814:F
809:G
805:G
797:G
793:G
785:)
783:F
781:(
770:F
765:.
763:G
759:x
738:)
735:1
732:(
724:)
721:x
718:(
709:]
706:)
703:x
700:(
695:G
691:C
687::
684:G
681:[
671:χ
666:C
662:F
656:.
649:C
645:F
640:g
632:m
628:χ
624:n
616:m
613:n
609:)
607:g
605:(
603:χ
597:.
595:H
591:H
584:G
562:0
551:.
548:F
544:G
539:F
531:F
527:G
502:ρ
485:,
481:}
477:)
474:1
471:(
458:=
455:)
452:g
449:(
433:G
427:g
423:{
384:ρ
382:χ
373:F
369:G
359:χ
354:ρ
346:χ
334:ρ
320:ρ
318:χ
289:)
286:)
283:g
280:(
274:(
265:=
262:)
259:g
256:(
232:F
228:G
223:ρ
221:χ
216:ρ
208:V
204:G
196:)
194:V
190:G
186:ρ
181:F
167:V
153:2
34:.
20:)
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