457:
71:
is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising conjecture, and it became known as the
Frobenius determinant theorem.
160:
301:
309:
194:
487:
241:
507:
214:
101:
648:
569:
684:
551:
674:
632:
25:
106:
246:
689:
679:
583:
452:{\displaystyle \det X_{G}=\prod _{j=1}^{r}P_{j}(x_{g_{1}},x_{g_{2}},\dots ,x_{g_{n}})^{\deg P_{j}}}
40:
165:
644:
565:
541:
601:
589:
555:
21:
658:
613:
579:
465:
219:
654:
640:
609:
605:
575:
492:
199:
86:
668:
593:
81:
560:
545:
60:
621:
547:
Pioneers of
Representation Theory: Frobenius, Burnside, Schur, and Brauer
489:'s are pairwise non-proportional irreducible polynomials and
495:
468:
312:
249:
222:
202:
168:
109:
89:
20:was a conjecture made in 1896 by the mathematician
501:
481:
451:
295:
235:
208:
188:
154:
95:
39:If one takes the multiplication table of a finite
313:
63:, then the determinant factors as a product of
8:
550:, History of Mathematics, Providence, R.I.:
598:Gesammelte mathematische Werke. Bände I–III
509:is the number of conjugacy classes of
637:Gesammelte Abhandlungen. Bände I, II, III
559:
494:
473:
467:
441:
430:
418:
413:
392:
387:
372:
367:
354:
344:
333:
320:
311:
285:
275:
270:
254:
248:
227:
221:
201:
178:
173:
167:
146:
127:
114:
108:
88:
155:{\displaystyle g_{1},g_{2},\dots ,g_{n}}
29:
529:
522:
33:
296:{\displaystyle a_{ij}=x_{g_{i}g_{j}}}
7:
600:, New York: Chelsea Publishing Co.,
622:"Lectures on Representation Theory"
196:be associated with each element of
32:), with an English translation in (
14:
67:irreducible polynomials, where
427:
360:
1:
561:10.1090/S0273-0979-00-00867-3
552:American Mathematical Society
59:, and subsequently takes the
18:Frobenius determinant theorem
635:(1968), Serre, J.-P. (ed.),
706:
633:Frobenius, Ferdinand Georg
592:(1968) , Fricke, Robert;
189:{\displaystyle x_{g_{i}}}
28:about it (reproduced in (
685:Theorems in group theory
46:and replaces each entry
24:, who wrote a letter to
620:Etingof, Pavel (2005).
596:; Ore, öystein (eds.),
503:
483:
453:
349:
297:
237:
210:
190:
156:
97:
504:
484:
482:{\displaystyle P_{j}}
454:
329:
298:
238:
236:{\displaystyle X_{G}}
211:
191:
157:
98:
639:, Berlin, New York:
493:
466:
310:
247:
220:
216:. Define the matrix
200:
166:
107:
87:
16:In mathematics, the
675:Theorems in algebra
542:Curtis, Charles W.
499:
479:
449:
293:
233:
206:
186:
152:
93:
50:with the variable
650:978-3-540-04120-7
590:Dedekind, Richard
571:978-0-8218-2677-5
502:{\displaystyle r}
209:{\displaystyle G}
96:{\displaystyle G}
697:
661:
628:
626:
616:
582:
563:
533:
527:
508:
506:
505:
500:
488:
486:
485:
480:
478:
477:
458:
456:
455:
450:
448:
447:
446:
445:
425:
424:
423:
422:
399:
398:
397:
396:
379:
378:
377:
376:
359:
358:
348:
343:
325:
324:
302:
300:
299:
294:
292:
291:
290:
289:
280:
279:
262:
261:
242:
240:
239:
234:
232:
231:
215:
213:
212:
207:
195:
193:
192:
187:
185:
184:
183:
182:
161:
159:
158:
153:
151:
150:
132:
131:
119:
118:
102:
100:
99:
94:
76:Formal statement
36:, p. 51)).
22:Richard Dedekind
705:
704:
700:
699:
698:
696:
695:
694:
665:
664:
651:
641:Springer-Verlag
631:
624:
619:
588:
572:
540:
537:
536:
528:
524:
519:
491:
490:
469:
464:
463:
437:
426:
414:
409:
388:
383:
368:
363:
350:
316:
308:
307:
281:
271:
266:
250:
245:
244:
223:
218:
217:
198:
197:
174:
169:
164:
163:
142:
123:
110:
105:
104:
85:
84:
78:
58:
26:F. G. Frobenius
12:
11:
5:
703:
701:
693:
692:
687:
682:
677:
667:
666:
663:
662:
649:
629:
617:
586:
570:
535:
534:
532:, Theorem 5.4.
521:
520:
518:
515:
498:
476:
472:
460:
459:
444:
440:
436:
433:
429:
421:
417:
412:
408:
405:
402:
395:
391:
386:
382:
375:
371:
366:
362:
357:
353:
347:
342:
339:
336:
332:
328:
323:
319:
315:
288:
284:
278:
274:
269:
265:
260:
257:
253:
230:
226:
205:
181:
177:
172:
149:
145:
141:
138:
135:
130:
126:
122:
117:
113:
103:have elements
92:
77:
74:
54:
13:
10:
9:
6:
4:
3:
2:
702:
691:
690:Matrix theory
688:
686:
683:
681:
678:
676:
673:
672:
670:
660:
656:
652:
646:
642:
638:
634:
630:
623:
618:
615:
611:
607:
603:
599:
595:
594:Noether, Emmy
591:
587:
585:
581:
577:
573:
567:
562:
557:
553:
549:
548:
543:
539:
538:
531:
526:
523:
516:
514:
512:
496:
474:
470:
442:
438:
434:
431:
419:
415:
410:
406:
403:
400:
393:
389:
384:
380:
373:
369:
364:
355:
351:
345:
340:
337:
334:
330:
326:
321:
317:
306:
305:
304:
286:
282:
276:
272:
267:
263:
258:
255:
251:
243:with entries
228:
224:
203:
179:
175:
170:
147:
143:
139:
136:
133:
128:
124:
120:
115:
111:
90:
83:
75:
73:
70:
66:
62:
57:
53:
49:
45:
42:
37:
35:
31:
30:Dedekind 1968
27:
23:
19:
680:Determinants
636:
597:
546:
530:Etingof 2005
525:
510:
461:
82:finite group
79:
68:
64:
55:
51:
47:
43:
38:
17:
15:
61:determinant
34:Curtis 2003
669:Categories
606:56.0024.05
517:References
462:where the
162:, and let
435:
404:…
331:∏
137:…
544:(2003),
659:0235974
614:0237282
580:1715145
303:. Then
657:
647:
612:
604:
584:Review
578:
568:
80:Let a
625:(PDF)
41:group
645:ISBN
566:ISBN
602:JFM
556:doi
432:deg
314:det
671::
655:MR
653:,
643:,
610:MR
608:,
576:MR
574:,
564:,
554:,
513:.
627:.
558::
511:G
497:r
475:j
471:P
443:j
439:P
428:)
420:n
416:g
411:x
407:,
401:,
394:2
390:g
385:x
381:,
374:1
370:g
365:x
361:(
356:j
352:P
346:r
341:1
338:=
335:j
327:=
322:G
318:X
287:j
283:g
277:i
273:g
268:x
264:=
259:j
256:i
252:a
229:G
225:X
204:G
180:i
176:g
171:x
148:n
144:g
140:,
134:,
129:2
125:g
121:,
116:1
112:g
91:G
69:n
65:n
56:g
52:x
48:g
44:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.