818:
257:
711:
529:
547:
157:
1017:
884:
923:
856:
59:
425:
642:
745:
1054:
334:
557:
186:
674:
532:
859:
736:
452:
1046:
572:
357:
93:
968:
536:
80:
926:
660:
591:
1089:
865:
732:
341:
318:
167:
893:
826:
69:
564:
is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a
560:
asks whether the ring of invariants is finitely generated or not (the answer is affirmative if
1050:
938:
583:
352:. Geometrically, the rings of invariants are the coordinate rings of (affine or projective)
349:
1064:
37:
1060:
599:
436:
428:
326:
369:
627:
1083:
310:
565:
353:
299:
170:
62:
17:
729:
714:
649:
645:
668:
813:{\displaystyle \mathbb {C} ^{G}\to \operatorname {H} ^{2*}(M;\mathbb {C} )}
722:
614:
76:
28:
598:
is a finitely generated algebra. The answer is negative for some
294:. In particular, the fixed-point subring of an automorphism
1045:, Cambridge Studies in Advanced Mathematics, vol. 81,
290:
form the ring of invariants under the group generated by
449:
by permuting the variables. Then the ring of invariants
356:
and they play fundamental roles in the constructions in
252:{\displaystyle R^{G}=\{r\in R\mid g\cdot r=r,\,g\in G\}}
1075:, Lecture Notes in Mathematics, vol. 585, Springer
706:{\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}
325:
is a group of field automorphisms, the fixed ring is a
971:
896:
868:
829:
748:
677:
630:
455:
372:
189:
96:
40:
1041:Mukai, Shigeru; Oxbury, W. M. (8 September 2003) ,
1011:
917:
878:
850:
812:
705:
636:
523:
419:
251:
151:
53:
613:be the symmetric algebra of a finite-dimensional
524:{\displaystyle R^{G}=k^{\operatorname {S} _{n}}}
8:
246:
203:
152:{\displaystyle R^{f}=\{r\in R\mid f(r)=r\}.}
143:
110:
1012:{\displaystyle \prod _{g\in G}(t-g\cdot r)}
976:
970:
906:
905:
898:
897:
895:
870:
869:
867:
839:
838:
831:
830:
828:
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802:
781:
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758:
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749:
747:
679:
678:
676:
629:
513:
508:
498:
479:
460:
454:
408:
389:
371:
236:
194:
188:
101:
95:
45:
39:
1043:An Introduction to Invariants and Moduli
950:
548:fundamental theorem of invariant theory
624:is a reflection group if and only if
7:
335:Fundamental theorem of Galois theory
907:
871:
840:
759:
680:
778:
510:
286:that are fixed by the elements of
25:
298:is the ring of invariants of the
348:is a central object of study in
879:{\displaystyle {\mathfrak {g}}}
333:of the automorphism group; see
1006:
988:
912:
902:
845:
835:
807:
793:
774:
765:
754:
700:
694:
505:
472:
414:
382:
134:
128:
1:
533:ring of symmetric polynomials
278:is a set of automorphisms of
918:{\displaystyle \mathbb {C} }
860:ring of polynomial functions
851:{\displaystyle \mathbb {C} }
558:Hilbert's fourteenth problem
550:describes the generators of
266:or, more traditionally, the
1071:Springer, Tonny A. (1977),
1019:is a monic polynomial over
1106:
1047:Cambridge University Press
573:finitely generated algebra
358:geometric invariant theory
537:reductive algebraic group
737:Chern–Weil homomorphism
656:(Chevalley's theorem).
609:be a finite group. Let
1013:
927:adjoint representation
919:
880:
852:
814:
717:, then each principal
707:
638:
525:
421:
253:
177:, then the subring of
153:
55:
1014:
920:
881:
853:
815:
708:
661:differential geometry
639:
526:
422:
254:
154:
56:
54:{\displaystyle R^{f}}
1027:as one of its roots.
969:
894:
866:
827:
746:
733:algebra homomorphism
675:
628:
453:
370:
342:module of covariants
187:
94:
38:
420:{\displaystyle R=k}
162:More generally, if
33:fixed-point subring
1009:
987:
915:
876:
848:
810:
703:
634:
521:
417:
346:ring of invariants
282:, the elements of
268:ring of invariants
249:
149:
51:
18:Ring of invariants
1056:978-0-521-80906-1
972:
965:, the polynomial
939:Character variety
637:{\displaystyle S}
16:(Redirected from
1097:
1076:
1073:Invariant theory
1067:
1028:
1018:
1016:
1015:
1010:
986:
955:
924:
922:
921:
916:
911:
910:
901:
885:
883:
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877:
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857:
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854:
849:
844:
843:
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819:
817:
816:
811:
806:
789:
788:
773:
772:
763:
762:
753:
712:
710:
709:
704:
684:
683:
643:
641:
640:
635:
600:unipotent groups
592:Artin–Tate lemma
530:
528:
527:
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520:
519:
518:
517:
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502:
484:
483:
465:
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350:invariant theory
273:
258:
256:
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198:
158:
156:
155:
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106:
105:
60:
58:
57:
52:
50:
49:
21:
1105:
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1100:
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1080:
1079:
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1057:
1040:
1037:
1032:
1031:
967:
966:
956:
952:
947:
935:
892:
891:
864:
863:
825:
824:
777:
764:
744:
743:
673:
672:
626:
625:
509:
504:
494:
475:
456:
451:
450:
444:
437:symmetric group
435:variables. The
429:polynomial ring
404:
385:
368:
367:
271:
190:
185:
184:
97:
92:
91:
41:
36:
35:
23:
22:
15:
12:
11:
5:
1103:
1101:
1093:
1092:
1082:
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1078:
1077:
1068:
1055:
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1033:
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1029:
1008:
1005:
1002:
999:
996:
993:
990:
985:
982:
979:
975:
949:
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942:
941:
934:
931:
914:
909:
904:
900:
873:
847:
842:
837:
833:
821:
820:
809:
805:
801:
798:
795:
792:
787:
784:
780:
776:
771:
767:
761:
756:
752:
702:
699:
696:
693:
690:
687:
682:
633:
516:
512:
507:
501:
497:
493:
490:
487:
482:
478:
474:
471:
468:
463:
459:
440:
416:
411:
407:
403:
400:
397:
392:
388:
384:
381:
378:
375:
262:is called the
260:
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248:
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48:
44:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1102:
1091:
1088:
1087:
1085:
1074:
1069:
1066:
1062:
1058:
1052:
1048:
1044:
1039:
1038:
1034:
1026:
1022:
1003:
1000:
997:
994:
991:
983:
980:
977:
973:
964:
960:
954:
951:
944:
940:
937:
936:
932:
930:
928:
889:
861:
799:
796:
790:
785:
782:
769:
742:
741:
740:
738:
734:
731:
728:determines a
727:
724:
721:-bundle on a
720:
716:
697:
691:
688:
685:
670:
666:
662:
657:
655:
651:
647:
631:
623:
619:
617:
612:
608:
603:
601:
597:
593:
589:
585:
581:
577:
574:
570:
567:
563:
559:
555:
553:
549:
545:
541:
538:
534:
514:
499:
495:
491:
488:
485:
480:
476:
469:
466:
461:
457:
448:
443:
438:
434:
430:
409:
405:
401:
398:
395:
390:
386:
379:
376:
373:
365:
361:
359:
355:
354:GIT quotients
351:
347:
343:
340:Along with a
338:
336:
332:
328:
324:
320:
316:
312:
311:Galois theory
307:
305:
302:generated by
301:
297:
293:
289:
285:
281:
277:
269:
265:
264:fixed subring
243:
240:
237:
233:
230:
227:
224:
221:
218:
215:
212:
209:
206:
200:
195:
191:
183:
182:
181:
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176:
172:
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165:
146:
140:
137:
131:
125:
122:
119:
116:
113:
107:
102:
98:
90:
89:
88:
86:
82:
78:
74:
71:
67:
64:
46:
42:
34:
30:
19:
1072:
1042:
1024:
1020:
962:
958:
953:
887:
822:
735:(called the
725:
718:
664:
658:
653:
621:
615:
610:
606:
604:
595:
587:
579:
575:
571:acting on a
568:
566:finite group
561:
556:
551:
543:
539:
446:
441:
432:
363:
362:
345:
339:
330:
322:
314:
308:
303:
300:cyclic group
295:
291:
287:
283:
279:
275:
267:
263:
261:
178:
174:
163:
161:
84:
81:fixed points
72:
65:
63:automorphism
32:
26:
1090:Ring theory
715:Lie algebra
648:(of finite
646:free module
546:, then the
331:fixed field
329:called the
87:, that is,
1035:References
1001:⋅
995:−
981:∈
974:∏
791:
786:∗
775:→
692:
669:Lie group
489:…
399:…
241:∈
222:⋅
216:∣
210:∈
123:∣
117:∈
1084:Category
1023:and has
933:See also
890:acts on
723:manifold
594:implies
584:integral
578:: since
542:acts on
445:acts on
327:subfield
1065:2004218
858:is the
652:) over
620:. Then
618:-module
535:. If a
531:is the
364:Example
313:, when
79:of the
77:subring
75:is the
29:algebra
1063:
1053:
957:Given
823:where
730:graded
590:, the
366:: Let
344:, the
270:under
171:acting
61:of an
31:, the
945:Notes
667:is a
663:, if
644:is a
586:over
427:be a
319:field
317:is a
274:. If
168:group
166:is a
68:of a
1051:ISBN
886:and
713:its
671:and
650:rank
605:Let
321:and
70:ring
961:in
925:by
862:on
689:Lie
659:In
582:is
431:in
309:In
173:on
83:of
27:In
1086::
1061:MR
1059:,
1049:,
929:.
739:)
602:.
554:.
360:.
337:.
306:.
1025:r
1021:R
1007:)
1004:r
998:g
992:t
989:(
984:G
978:g
963:R
959:r
913:]
908:g
903:[
899:C
888:G
872:g
846:]
841:g
836:[
832:C
808:)
804:C
800:;
797:M
794:(
783:2
779:H
770:G
766:]
760:g
755:[
751:C
726:M
719:G
701:)
698:G
695:(
686:=
681:g
665:G
654:S
632:S
622:G
616:G
611:S
607:G
596:R
588:R
580:R
576:R
569:G
562:G
552:R
544:R
540:G
515:n
511:S
506:]
500:n
496:x
492:,
486:,
481:1
477:x
473:[
470:k
467:=
462:G
458:R
447:R
442:n
439:S
433:n
415:]
410:n
406:x
402:,
396:,
391:1
387:x
383:[
380:k
377:=
374:R
323:G
315:R
304:f
296:f
292:S
288:S
284:R
280:R
276:S
272:G
247:}
244:G
238:g
234:,
231:r
228:=
225:r
219:g
213:R
207:r
204:{
201:=
196:G
192:R
179:R
175:R
164:G
147:.
144:}
141:r
138:=
135:)
132:r
129:(
126:f
120:R
114:r
111:{
108:=
103:f
99:R
85:f
73:R
66:f
47:f
43:R
20:)
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