198:, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors.
513:
The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian), and if the indicator is −1, the representation is quaternionic.
508:
409:
254:
96:
296:
138:
710:
670:
202:
205:, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a
436:
729:
642:
654:
302:
355:
527:
214:
646:
329:
206:
538:
312:
of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant
218:
61:
24:
32:
227:
69:
591:
309:
164:
583:
574:
265:
107:
706:
684:
666:
313:
658:
595:
579:
342:
680:
676:
542:
523:
176:
163:
is a representation on a complex vector space with an antilinear equivariant map given by
621:
form, so the significance of zero indicator is that there is no invariant nondegenerate
699:
531:
317:
54:
46:
305:
of real and quaternionic representations is neither real nor quaternionic in general.
723:
335:
541:
on odd-dimensional spaces are real, since they all appear as subrepresentations of
423:
39:
36:
20:
320:. Such representations are sometimes said to be complex or (pseudo-)hermitian.
662:
553:
188:
184:
58:
688:
587:
308:
A representation on a complex vector space can also be isomorphic to the
195:
526:
are real (and in fact rational), since we can build a complete set of
657:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
549:
201:
A real representation on a complex vector space is isomorphic to its
545:
of copies of the fundamental representation, which is real.
341:) for reality of irreducible representations in terms of
503:{\displaystyle {1 \over |G|}\sum _{g\in G}\chi (g^{2}).}
439:
358:
268:
230:
110:
72:
578:
8 is known in mathematics not only in the theory of
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502:
403:
290:
248:
132:
90:
548:Further examples of real representations are the
404:{\displaystyle \int _{g\in G}\chi (g^{2})\,d\mu }
16:Type of representation in representation theory
143:The two viewpoints are equivalent because if
45:, but it can also mean a representation on a
8:
430:) = 1. For a finite group, this is given by
418:is the character of the representation and
209:. An irreducible pseudoreal representation
147:is a real vector space acted on by a group
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440:
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394:
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363:
357:
273:
267:
229:
115:
109:
71:
701:Linear Representations of Finite Groups
606:
171:is such a complex representation, then
221:, i.e., an antilinear equivariant map
651:Representation theory. A first course
7:
617:of a compact group has an invariant
14:
203:complex conjugate representation
572:= 1, 2, 3 ... This periodicity
494:
481:
455:
447:
391:
378:
249:{\displaystyle j\colon V\to V}
240:
91:{\displaystyle j\colon V\to V}
82:
1:
655:Graduate Texts in Mathematics
697:Serre, Jean-Pierre (1977),
613:Any complex representation
537:All representations of the
528:irreducible representations
215:quaternionic representation
746:
522:All representation of the
327:
663:10.1007/978-1-4612-0979-9
347:Frobenius-Schur indicator
330:Frobenius-Schur indicator
324:Frobenius-Schur indicator
291:{\displaystyle j^{2}=-1.}
217:: it admits an invariant
207:pseudoreal representation
133:{\displaystyle j^{2}=+1.}
175:can be recovered as the
552:representations of the
504:
405:
292:
250:
219:quaternionic structure
134:
92:
730:Representation theory
505:
406:
293:
251:
135:
93:
25:representation theory
437:
356:
266:
228:
108:
70:
705:, Springer-Verlag,
592:spin representation
310:dual representation
165:complex conjugation
29:real representation
584:algebraic topology
568:+1 dimensions for
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477:
401:
288:
246:
130:
88:
53:with an invariant
712:978-0-387-90190-9
672:978-0-387-97495-8
580:Clifford algebras
462:
460:
334:A criterion (for
314:sesquilinear form
213:is necessarily a
167:. Conversely, if
737:
715:
704:
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623:complex bilinear
611:
596:Bott periodicity
524:symmetric groups
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345:is based on the
343:character theory
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259:which satisfies
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120:
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101:which satisfies
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643:Fulton, William
641:
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543:tensor products
539:rotation groups
520:
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177:fixed point set
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62:equivariant map
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12:
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5:
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582:, but also in
532:Young tableaux
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336:compact groups
328:Main article:
325:
322:
318:hermitian form
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99:
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75:
55:real structure
33:representation
15:
13:
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6:
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3:
2:
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731:
728:
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451:
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127:
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66:
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49:vector space
48:
44:
41:
38:
34:
31:is usually a
30:
26:
22:
700:
650:
626:
622:
618:
614:
609:
573:
569:
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561:
557:
547:
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521:
512:
427:
424:Haar measure
419:
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346:
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307:
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210:
200:
193:
180:
172:
168:
160:
156:
152:
151:(say), then
148:
144:
142:
100:
50:
42:
40:vector space
28:
21:mathematical
18:
647:Harris, Joe
560:−1, 8
554:spin groups
349:defined by
57:, i.e., an
636:References
303:direct sum
189:eigenvalue
185:eigenspace
59:antilinear
689:246650103
619:hermitian
588:KO-theory
479:χ
471:∈
464:∑
399:μ
376:χ
368:∈
361:∫
316:, e.g. a
283:−
241:→
235::
83:→
77::
23:field of
724:Category
649:(1991).
625:form on
518:Examples
681:1153249
564:, and 8
426:with μ(
422:is the
196:physics
159:⊗
47:complex
19:In the
709:
687:
679:
669:
590:; see
575:modulo
550:spinor
530:using
416:χ
414:where
602:Notes
586:, in
187:with
183:(the
35:on a
707:ISBN
685:OCLC
667:ISBN
594:and
556:in 8
191:1).
37:real
659:doi
194:In
179:of
726::
683:.
677:MR
675:.
665:.
653:.
645:;
598:.
534:.
301:A
286:1.
155:=
128:1.
27:a
716:.
693:.
691:.
661::
629:.
627:V
615:V
570:k
566:k
562:k
558:k
498:.
495:)
490:2
486:g
482:(
474:G
468:g
456:|
452:G
448:|
443:1
428:G
420:μ
396:d
392:)
387:2
383:g
379:(
371:G
365:g
339:G
280:=
275:2
271:j
244:V
238:V
232:j
211:V
181:j
173:U
169:V
161:C
157:U
153:V
149:G
145:U
125:+
122:=
117:2
113:j
86:V
80:V
74:j
51:V
43:U
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