807:
246:
700:
518:
146:
1006:
873:
912:
845:
536:
48:
414:
631:
734:
175:
1043:
663:
323:
441:
82:
957:
546:
521:
848:
725:
1035:
561:
346:
854:
525:
69:
882:
815:
553:
is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a
915:
549:
asks whether the ring of invariants is finitely generated or not (the answer is affirmative if
649:
341:. Geometrically, the rings of invariants are the coordinate rings of (affine or projective)
1078:
1053:
721:
580:
330:
26:
8:
358:
307:
156:
616:
58:
1039:
927:
638:
572:
338:
1049:
588:
425:
417:
315:
1072:
299:
802:{\displaystyle \mathbb {C} ^{G}\to \operatorname {H} ^{2*}(M;\mathbb {C} )}
554:
342:
288:
159:
51:
718:
703:
634:
657:
711:
603:
587:
is a finitely generated algebra. The answer is negative for some
65:
17:
283:. In particular, the fixed-point subring of an automorphism
1034:, Cambridge Studies in Advanced Mathematics, vol. 81,
279:
form the ring of invariants under the group generated by
438:
by permuting the variables. Then the ring of invariants
345:
and they play fundamental roles in the constructions in
241:{\displaystyle R^{G}=\{r\in R\mid g\cdot r=r,\,g\in G\}}
1064:, Lecture Notes in Mathematics, vol. 585, Springer
695:{\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}
314:
is a group of field automorphisms, the fixed ring is a
960:
885:
857:
818:
737:
666:
619:
444:
361:
178:
85:
29:
1030:Mukai, Shigeru; Oxbury, W. M. (8 September 2003) ,
1000:
906:
867:
839:
801:
694:
625:
512:
408:
240:
140:
42:
602:be the symmetric algebra of a finite-dimensional
1070:
513:{\displaystyle R^{G}=k^{\operatorname {S} _{n}}}
235:
192:
141:{\displaystyle R^{f}=\{r\in R\mid f(r)=r\}.}
132:
99:
1029:
1001:{\displaystyle \prod _{g\in G}(t-g\cdot r)}
887:
820:
792:
739:
225:
1059:
1032:An Introduction to Invariants and Moduli
537:fundamental theorem of invariant theory
1071:
613:is a reflection group if and only if
324:Fundamental theorem of Galois theory
896:
860:
829:
748:
669:
13:
767:
499:
275:that are fixed by the elements of
14:
1090:
287:is the ring of invariants of the
337:is a central object of study in
868:{\displaystyle {\mathfrak {g}}}
322:of the automorphism group; see
995:
977:
940:
901:
891:
834:
824:
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782:
763:
754:
743:
689:
683:
494:
461:
403:
371:
123:
117:
1:
1023:
522:ring of symmetric polynomials
267:is a set of automorphisms of
907:{\displaystyle \mathbb {C} }
849:ring of polynomial functions
840:{\displaystyle \mathbb {C} }
547:Hilbert's fourteenth problem
539:describes the generators of
255:or, more traditionally, the
7:
1060:Springer, Tonny A. (1977),
1008:is a monic polynomial over
921:
10:
1095:
1036:Cambridge University Press
562:finitely generated algebra
347:geometric invariant theory
526:reductive algebraic group
933:
726:Chern–Weil homomorphism
645:(Chevalley's theorem).
598:be a finite group. Let
1002:
916:adjoint representation
908:
869:
841:
803:
706:, then each principal
696:
627:
514:
410:
242:
166:, then the subring of
142:
44:
1003:
909:
870:
842:
804:
697:
650:differential geometry
628:
515:
411:
243:
143:
45:
43:{\displaystyle R^{f}}
1016:as one of its roots.
958:
883:
855:
816:
735:
722:algebra homomorphism
664:
617:
442:
359:
331:module of covariants
176:
83:
27:
409:{\displaystyle R=k}
151:More generally, if
22:fixed-point subring
998:
976:
904:
865:
837:
799:
692:
623:
510:
406:
335:ring of invariants
271:, the elements of
257:ring of invariants
238:
138:
40:
1045:978-0-521-80906-1
961:
954:, the polynomial
928:Character variety
626:{\displaystyle S}
1086:
1065:
1062:Invariant theory
1056:
1017:
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589:unipotent groups
581:Artin–Tate lemma
519:
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339:invariant theory
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426:symmetric group
424:variables. The
418:polynomial ring
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260:
183:
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90:
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34:
30:
28:
25:
24:
12:
11:
5:
1092:
1082:
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1067:
1066:
1057:
1044:
1025:
1022:
1019:
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994:
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964:
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923:
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898:
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862:
836:
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826:
822:
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787:
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773:
769:
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691:
688:
685:
682:
679:
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671:
622:
505:
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479:
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448:
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251:is called the
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107:
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37:
33:
9:
6:
4:
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2:
1091:
1080:
1077:
1076:
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1055:
1051:
1047:
1041:
1037:
1033:
1028:
1027:
1015:
1011:
992:
989:
986:
983:
980:
972:
969:
966:
962:
953:
949:
943:
939:
929:
926:
925:
919:
917:
878:
850:
788:
785:
779:
774:
771:
758:
731:
730:
729:
727:
723:
720:
717:determines a
716:
713:
710:-bundle on a
709:
705:
686:
680:
677:
674:
659:
655:
651:
646:
644:
640:
636:
620:
612:
608:
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582:
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552:
548:
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527:
523:
503:
488:
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477:
474:
469:
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458:
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437:
432:
427:
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398:
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384:
379:
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368:
365:
362:
354:
350:
348:
344:
343:GIT quotients
340:
336:
332:
329:Along with a
327:
325:
321:
317:
313:
309:
305:
301:
300:Galois theory
296:
294:
291:generated by
290:
286:
282:
278:
274:
270:
266:
258:
254:
253:fixed subring
232:
229:
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135:
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96:
91:
87:
79:
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71:
67:
63:
60:
56:
53:
35:
31:
23:
19:
1061:
1031:
1013:
1009:
951:
947:
942:
876:
811:
724:(called the
714:
707:
653:
647:
642:
610:
604:
599:
595:
593:
584:
576:
568:
564:
560:acting on a
557:
555:finite group
550:
545:
540:
532:
528:
435:
430:
421:
352:
351:
334:
328:
319:
311:
303:
297:
292:
289:cyclic group
284:
280:
276:
272:
268:
264:
256:
252:
250:
167:
163:
152:
150:
73:
70:fixed points
61:
54:
52:automorphism
21:
15:
1079:Ring theory
704:Lie algebra
637:(of finite
635:free module
535:, then the
320:fixed field
318:called the
76:, that is,
1024:References
990:⋅
984:−
970:∈
963:∏
780:
775:∗
764:→
681:
658:Lie group
478:…
388:…
230:∈
211:⋅
205:∣
199:∈
112:∣
106:∈
1073:Category
1012:and has
922:See also
879:acts on
712:manifold
583:implies
573:integral
567:: since
531:acts on
434:acts on
316:subfield
1054:2004218
847:is the
641:) over
609:. Then
607:-module
524:. If a
520:is the
353:Example
302:, when
68:of the
66:subring
64:is the
18:algebra
1052:
1042:
946:Given
812:where
719:graded
579:, the
355:: Let
333:, the
259:under
160:acting
50:of an
20:, the
934:Notes
656:is a
652:, if
633:is a
575:over
416:be a
308:field
306:is a
263:. If
157:group
155:is a
57:of a
1040:ISBN
875:and
702:its
660:and
639:rank
594:Let
310:and
59:ring
950:in
914:by
851:on
678:Lie
648:In
571:is
420:in
298:In
162:on
72:of
16:In
1075::
1050:MR
1048:,
1038:,
918:.
728:)
591:.
543:.
349:.
326:.
295:.
1014:r
1010:R
996:)
993:r
987:g
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978:(
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897:g
892:[
888:C
877:G
861:g
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830:g
825:[
821:C
797:)
793:C
789:;
786:M
783:(
772:2
768:H
759:G
755:]
749:g
744:[
740:C
715:M
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687:G
684:(
675:=
670:g
654:G
643:S
621:S
611:G
605:G
600:S
596:G
585:R
577:R
569:R
565:R
558:G
551:G
541:R
533:R
529:G
504:n
500:S
495:]
489:n
485:x
481:,
475:,
470:1
466:x
462:[
459:k
456:=
451:G
447:R
436:R
431:n
428:S
422:n
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399:n
395:x
391:,
385:,
380:1
376:x
372:[
369:k
366:=
363:R
312:G
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293:f
285:f
281:S
277:S
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261:G
236:}
233:G
227:g
223:,
220:r
217:=
214:r
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193:{
190:=
185:G
181:R
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133:}
130:r
127:=
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121:r
118:(
115:f
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