330:
578:
174:
or its opposite subalgebra. In the finite-dimensional these are exchanged by the longest element of the Weyl group, but this is no longer the case in infinite dimensions as there is no longest element.
471:
418:
246:
495:
949:
37:
121:λ of the highest weight vector λ under this action are the extremal weights, whose weight spaces are all 1-dimensional.
772:
896:
613:
423:
170:, except that one now has 2 cases as one can consider the submodules generated by either the Borel subalgebra
848:
Mehta, V. B.; Ramanathan, A. (1985), "Frobenius splitting and cohomology vanishing for
Schubert varieties",
57:
167:
850:
894:
Ramanan, S.; Ramanathan, A. (1985), "Projective normality of flag varieties and
Schubert varieties",
224:
gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques.
929:
875:
810:
646:
209:
913:
867:
829:
789:
748:
710:
679:
630:
325:{\displaystyle {\text{Ch}}(F(w\lambda ))=\Delta _{1}\Delta _{2}\cdots \Delta _{n}e^{\lambda }}
84:
56:, theorem 2), gives the characters of Demazure modules, and is a generalization of the
905:
859:
819:
781:
767:
738:
669:
622:
77:
41:
925:
887:
841:
801:
760:
722:
691:
642:
403:
921:
883:
837:
797:
756:
718:
698:
687:
657:
638:
49:
21:
166:
Demazure modules can be defined in a similar way for highest weight representations of
943:
933:
650:
60:. The dimension of a Demazure module is a polynomial in the highest weight, called a
573:{\displaystyle \Delta _{\alpha }(u)={\frac {u-s_{\alpha }\cdot u}{1-e^{-\alpha }}}}
785:
770:(1993), "The crystal base and Littelmann's refined Demazure character formula",
611:
Andersen, H. H. (1985), "Schubert varieties and
Demazure's character formula",
208:
gave a proof of
Demazure's character formula using the work on the geometry of
193:
107:
917:
871:
833:
793:
752:
714:
683:
634:
33:
824:
743:
674:
196:
pointed out that
Demazure's proof has a serious gap, as it depends on (
159:
then the
Demazure module is the whole of the irreducible representation
102:, and the highest weight space is 1-dimensional and is an eigenspace of
909:
879:
626:
863:
660:(1974a), "Désingularisation des variétés de Schubert généralisées",
420:
is the sum of fundamental weights and the dot action is defined by
369:
the corresponding element of the group ring of the weight lattice.
228:
proved a refined version of the
Demazure character formula that
36:
of a finite-dimensional representation generated by an extremal
808:
Littelmann, Peter (1995), "Crystal graphs and Young tableaux",
729:
Joseph, Anthony (1985), "On the
Demazure character formula",
151:
is trivial the
Demazure module is just 1-dimensional, and if
342:
is an element of the Weyl group, with reduced decomposition
498:
426:
406:
249:
132:
generated by the weight space of an extremal vector
731:
662:
Annales Scientifiques de l'École Normale Supérieure
200:, Proposition 11, section 2), which is false; see (
90:. An irreducible finite-dimensional representation
572:
465:
412:
324:
188:The Demazure character formula was introduced by (
213:
701:(1974b), "Une nouvelle formule des caractères",
217:
8:
380:λ)) is the character of the Demazure module
362:as a product of reflections of simple roots.
76:is a complex semisimple Lie algebra, with a
229:
823:
742:
673:
558:
534:
521:
503:
497:
466:{\displaystyle w\cdot u=w(u+\rho )-\rho }
425:
405:
316:
306:
293:
283:
250:
248:
225:
197:
189:
53:
29:
25:
480:for α a root is the endomorphism of the
232:conjectured (and proved in many cases).
205:
204:, section 4) for Kac's counterexample.
221:
201:
7:
155:is the element of maximal length of
703:Bulletin des Sciences Mathématiques
140:are parametrized by the Weyl group
500:
303:
290:
280:
240:The Demazure character formula is
98:splits as a sum of eigenspaces of
14:
136:λ, so the Demazure submodules of
147:There are two extreme cases: if
214:Ramanan & Ramanathan (1985)
515:
509:
454:
442:
273:
270:
261:
255:
1:
786:10.1215/S0012-7094-93-07131-1
218:Mehta & Ramanathan (1985)
40:space under the action of a
394:is the weight lattice, and
966:
365:λ is a lowest weight, and
179:Demazure character formula
46:Demazure character formula
773:Duke Mathematical Journal
124:A Demazure module is the
897:Inventiones Mathematicae
614:Inventiones Mathematicae
113:acts on the weights of
825:10.1006/jabr.1995.1175
574:
467:
414:
326:
58:Weyl character formula
950:Representation theory
851:Annals of Mathematics
575:
468:
415:
413:{\displaystyle \rho }
327:
117:, and the conjugates
496:
424:
404:
247:
744:10.24033/asens.1493
675:10.24033/asens.1261
62:Demazure polynomial
16:In mathematics, a
910:10.1007/BF01388970
811:Journal of Algebra
627:10.1007/BF01388527
592:for α the root of
570:
463:
410:
398:is its group ring.
322:
210:Schubert varieties
168:Kac–Moody algebras
854:, Second Series,
768:Kashiwara, Masaki
568:
253:
230:Littelmann (1995)
85:Cartan subalgebra
957:
936:
890:
844:
827:
804:
763:
746:
725:
699:Demazure, Michel
694:
677:
658:Demazure, Michel
653:
579:
577:
576:
571:
569:
567:
566:
565:
546:
539:
538:
522:
508:
507:
472:
470:
469:
464:
419:
417:
416:
411:
331:
329:
328:
323:
321:
320:
311:
310:
298:
297:
288:
287:
254:
251:
226:Kashiwara (1993)
78:Borel subalgebra
68:Demazure modules
48:, introduced by
42:Borel subalgebra
20:, introduced by
965:
964:
960:
959:
958:
956:
955:
954:
940:
939:
893:
864:10.2307/1971368
847:
807:
766:
728:
697:
656:
610:
607:
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591:
587:
554:
547:
530:
523:
499:
494:
493:
479:
422:
421:
402:
401:
361:
352:
312:
302:
289:
279:
245:
244:
238:
206:Andersen (1985)
186:
181:
70:
18:Demazure module
12:
11:
5:
963:
961:
953:
952:
942:
941:
938:
937:
904:(2): 217–224,
891:
845:
805:
780:(3): 839–858,
764:
737:(3): 389–419,
726:
709:(3): 163–172,
695:
654:
621:(3): 611–618,
606:
603:
602:
601:
596:
589:
583:
580:
564:
561:
557:
553:
550:
545:
542:
537:
533:
529:
526:
520:
517:
514:
511:
506:
502:
490:
489:
477:
474:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
409:
399:
389:
370:
363:
357:
350:
333:
332:
319:
315:
309:
305:
301:
296:
292:
286:
282:
278:
275:
272:
269:
266:
263:
260:
257:
237:
234:
198:Demazure 1974a
192:, theorem 2).
190:Demazure 1974b
185:
182:
180:
177:
128:-submodule of
69:
66:
13:
10:
9:
6:
4:
3:
2:
962:
951:
948:
947:
945:
935:
931:
927:
923:
919:
915:
911:
907:
903:
899:
898:
892:
889:
885:
881:
877:
873:
869:
865:
861:
857:
853:
852:
846:
843:
839:
835:
831:
826:
821:
817:
813:
812:
806:
803:
799:
795:
791:
787:
783:
779:
775:
774:
769:
765:
762:
758:
754:
750:
745:
740:
736:
732:
727:
724:
720:
716:
712:
708:
704:
700:
696:
693:
689:
685:
681:
676:
671:
667:
663:
659:
655:
652:
648:
644:
640:
636:
632:
628:
624:
620:
616:
615:
609:
608:
604:
599:
595:
586:
581:
562:
559:
555:
551:
548:
543:
540:
535:
531:
527:
524:
518:
512:
504:
492:
491:
487:
483:
475:
460:
457:
451:
448:
445:
439:
436:
433:
430:
427:
407:
400:
397:
393:
390:
387:
383:
379:
375:
371:
368:
364:
360:
356:
349:
346: =
345:
341:
338:
337:
336:
317:
313:
307:
299:
294:
284:
276:
267:
264:
258:
243:
242:
241:
235:
233:
231:
227:
223:
222:Joseph (1985)
219:
215:
211:
207:
203:
199:
195:
191:
183:
178:
176:
173:
169:
164:
162:
158:
154:
150:
145:
143:
139:
135:
131:
127:
122:
120:
116:
112:
109:
105:
101:
97:
93:
89:
86:
83:containing a
82:
79:
75:
72:Suppose that
67:
65:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
901:
895:
858:(1): 27–40,
855:
849:
818:(1): 65–87,
815:
809:
777:
771:
734:
730:
706:
705:, 2e SĂ©rie,
702:
665:
661:
618:
612:
597:
593:
584:
485:
481:
395:
391:
385:
381:
377:
373:
366:
358:
354:
347:
343:
339:
334:
239:
187:
171:
165:
160:
156:
152:
148:
146:
141:
137:
133:
129:
125:
123:
118:
114:
110:
103:
99:
95:
91:
87:
80:
73:
71:
61:
45:
17:
15:
733:, SĂ©rie 4,
664:, SĂ©rie 4,
202:Joseph 1985
605:References
488:defined by
194:Victor Kac
108:Weyl group
934:123105737
918:0020-9910
872:0003-486X
834:0021-8693
794:0012-7094
753:0012-9593
715:0007-4497
684:0012-9593
668:: 53–88,
651:121295084
635:0020-9910
563:α
560:−
552:−
541:⋅
536:α
528:−
505:α
501:Δ
461:ρ
458:−
452:ρ
431:⋅
408:ρ
318:λ
304:Δ
300:⋯
291:Δ
281:Δ
268:λ
236:Statement
34:submodule
944:Category
484:-module
50:Demazure
32:), is a
22:Demazure
926:0778124
888:0799251
880:1971368
842:1338967
802:1240605
761:0826100
723:0430001
692:0354697
643:0782239
184:History
52: (
24: (
932:
924:
916:
886:
878:
870:
840:
832:
800:
792:
759:
751:
721:
713:
690:
682:
649:
641:
633:
335:Here:
106:. The
44:. The
38:weight
930:S2CID
876:JSTOR
647:S2CID
582:and Δ
54:1974b
30:1974b
26:1974a
914:ISSN
868:ISSN
830:ISSN
790:ISSN
749:ISSN
711:ISSN
680:ISSN
631:ISSN
588:is Δ
216:and
906:doi
860:doi
856:122
820:doi
816:175
782:doi
739:doi
670:doi
623:doi
388:λ).
372:Ch(
353:...
220:.
212:by
94:of
946::
928:,
922:MR
920:,
912:,
902:79
900:,
884:MR
882:,
874:,
866:,
838:MR
836:,
828:,
814:,
798:MR
796:,
788:,
778:71
776:,
757:MR
755:,
747:,
735:18
719:MR
717:,
707:98
688:MR
686:,
678:,
645:,
639:MR
637:,
629:,
619:79
617:,
252:Ch
163:.
144:.
64:.
28:,
908::
862::
822::
784::
741::
672::
666:7
625::
598:j
594:s
590:α
585:j
556:e
549:1
544:u
532:s
525:u
519:=
516:)
513:u
510:(
486:Z
482:Z
478:α
476:Δ
473:.
455:)
449:+
446:u
443:(
440:w
437:=
434:u
428:w
396:Z
392:P
386:w
384:(
382:F
378:w
376:(
374:F
367:e
359:n
355:s
351:1
348:s
344:w
340:w
314:e
308:n
295:2
285:1
277:=
274:)
271:)
265:w
262:(
259:F
256:(
172:b
161:V
157:W
153:w
149:w
142:W
138:V
134:w
130:V
126:b
119:w
115:V
111:W
104:b
100:h
96:g
92:V
88:h
81:b
74:g
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