412:
677:
1188:
566:
82:
1237:
454:
716:
612:
253:
156:
1041:
1074:
1270:. (Co)homology of the Grassmannian, and more generally, of more general flag varieties, has a basis consisting of the (co)homology classes of Schubert varieties, or
1124:
950:
881:
977:
300:
1005:
831:
1208:
1094:
924:
904:
855:
800:
780:
757:
586:
273:
227:
207:
176:
122:
102:
308:
1519:
1487:
425:-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of
617:
1461:
1403:
1534:
1267:
1252:
1524:
1471:
1441:
1307:
1129:
1291:
1424:
1335:
1361:
1529:
1514:
1419:
734:
132:, whose elements satisfy conditions giving lower bounds to the dimensions of the intersections of its elements
1279:
125:
1263:
1256:
516:-axis, it can be rotated or tilted around any point on the axis, and this excess of possible motions makes
530:
46:
1448:. London Mathematical Society Student Texts. Vol. 35. Cambridge, U.K.: Cambridge University Press.
1303:
1414:
1213:
1356:
1283:
428:
1366:
1339:
1315:
179:
682:
1299:
28:
591:
232:
135:
1483:
1457:
1399:
1379:
1351:
1010:
803:
505:
36:
1046:
1449:
1430:
1391:
1319:
1295:
1275:
1099:
929:
860:
1497:
955:
278:
1493:
1479:
1446:
Young
Tableaux. With Applications to Representation Theory and Geometry, Chapts. 5 and 9.4
1311:
737:
982:
808:
1323:
1193:
1079:
909:
889:
840:
785:
765:
760:
742:
571:
258:
212:
192:
183:
161:
107:
87:
718:. (In the example above, this would mean requiring certain intersections of the line
1508:
1331:
1287:
1248:
834:
129:
40:
1262:
The algebras of regular functions on
Schubert varieties have deep significance in
1383:
1298:. The study continued in the 20th century as part of the general development of
418:
1247:
Schubert varieties form one of the most important and best studied classes of
884:
1453:
407:{\displaystyle X\ =\ \{w\subset V\mid \dim(w)=2,\,\dim(w\cap V_{2})\geq 1\}.}
17:
1274:. The study of the intersection theory on the Grassmannian was initiated by
1435:
1395:
1327:
1251:. A certain measure of singularity of Schubert varieties is provided by
568:
is defined by specifying the minimal dimension of intersection of a
672:{\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V}
1338:
in combinatorics in the 1980s, and Fulton and MacPherson in
182:, but most commonly this taken to be either the real or the
1437:
Mitt. Math. Gesellschaft
Hamburg, 1 (1889) pp. 134–155
614:
with each of the spaces in a fixed reference complete flag
275:
that intersect a fixed (reference) 2-dimensional subspace
1306:, but accelerated in the 1990s beginning with the work of
1183:{\displaystyle G/P=\mathbf {Gr} _{k}(\mathbf {C} ^{n})}
857:-orbits, which may be parametrized by certain elements
158:, with the elements of a specified complete flag. Here
1390:. Wiley Classics Library edition. Wiley-Interscience.
492:°. Since there are three degrees of freedom in moving
1216:
1196:
1132:
1102:
1082:
1049:
1013:
985:
958:
932:
912:
892:
863:
843:
811:
788:
768:
745:
685:
620:
594:
574:
533:
431:
311:
281:
261:
235:
215:
195:
164:
138:
110:
90:
49:
1342:
of singular algebraic varieties, also in the 1980s.
468:(not necessarily through the origin) which meet the
1231:
1202:
1182:
1118:
1088:
1068:
1035:
999:
971:
944:
918:
898:
875:
849:
825:
794:
774:
751:
710:
671:
606:
580:
560:
448:
406:
294:
267:
247:
221:
201:
170:
150:
116:
96:
76:
1255:, which encode their local Goresky–MacPherson
464:. This is isomorphic to the set of all lines
8:
1318:, following up on earlier investigations of
398:
324:
1282:in the 19th century under the heading of
1223:
1218:
1215:
1195:
1171:
1166:
1156:
1148:
1136:
1131:
1110:
1101:
1081:
1060:
1048:
1027:
1012:
989:
984:
963:
957:
931:
911:
891:
862:
842:
815:
810:
787:
767:
744:
696:
684:
657:
638:
625:
619:
593:
573:
543:
535:
532:
484:in space (while keeping contact with the
433:
432:
430:
383:
363:
310:
286:
280:
260:
234:
214:
194:
163:
137:
128:. Like the Grassmannian, it is a kind of
109:
104:-dimensional subspaces of a vector space
89:
59:
51:
48:
178:may be a vector space over an arbitrary
1330:in representation theory in the 1970s,
1290:important enough to be included as the
527:More generally, a Schubert variety in
421:field, this can be pictured in usual
7:
1007:. The classical case corresponds to
979:and is called a Schubert variety in
733:In even greater generality, given a
561:{\displaystyle \mathbf {Gr} _{k}(V)}
77:{\displaystyle \mathbf {Gr} _{k}(V)}
782:and a standard parabolic subgroup
25:
1268:algebras with a straightening law
1096:th maximal parabolic subgroup of
488:-axis) corresponds to a curve in
1388:Principles of algebraic geometry
1232:{\displaystyle \mathbf {C} ^{n}}
1219:
1167:
1152:
1149:
926:-orbit associated to an element
539:
536:
55:
52:
500:-axis, rotating, and tilting),
449:{\displaystyle \mathbb {P} (V)}
1520:Topology of homogeneous spaces
1177:
1162:
555:
549:
443:
437:
389:
370:
351:
345:
71:
65:
1:
189:A typical example is the set
1249:singular algebraic varieties
837:, consists of finitely many
711:{\displaystyle \dim V_{j}=j}
504:is a three-dimensional real
1420:Encyclopedia of Mathematics
1253:Kazhdan–Lusztig polynomials
833:, which is an example of a
480:°, and continuously moving
456:, we obtain an open subset
1551:
1286:. This area was deemed by
607:{\displaystyle w\subset V}
476:corresponds to a point of
255:of a 4-dimensional space
248:{\displaystyle w\subset V}
151:{\displaystyle w\subset V}
496:(moving the point on the
1454:10.1017/CBO9780511626241
1413:A.L. Onishchik (2001) ,
1362:Bott–Samelson resolution
1036:{\displaystyle G=SL_{n}}
1535:Algebraic combinatorics
1264:algebraic combinatorics
1257:intersection cohomology
1190:is the Grassmannian of
1069:{\displaystyle P=P_{k}}
802:, it is known that the
229:-dimensional subspaces
1233:
1204:
1184:
1120:
1119:{\displaystyle SL_{n}}
1090:
1070:
1037:
1001:
973:
946:
945:{\displaystyle w\in W}
920:
900:
877:
876:{\displaystyle w\in W}
851:
827:
796:
776:
753:
712:
673:
608:
588:-dimensional subspace
582:
562:
472:-axis. Each such line
450:
408:
296:
269:
249:
223:
203:
172:
152:
118:
98:
78:
1525:Representation theory
1396:10.1002/9781118032527
1304:representation theory
1234:
1205:
1185:
1121:
1091:
1071:
1038:
1002:
974:
972:{\displaystyle X_{w}}
947:
921:
906:. The closure of the
901:
878:
852:
828:
797:
777:
754:
713:
674:
609:
583:
563:
451:
409:
297:
295:{\displaystyle V_{2}}
270:
250:
224:
204:
173:
153:
119:
99:
79:
1478:. Berlin, New York:
1357:Bruhat decomposition
1316:Schubert polynomials
1284:enumerative geometry
1266:and are examples of
1214:
1194:
1130:
1100:
1080:
1047:
1011:
983:
956:
930:
910:
890:
861:
841:
809:
786:
766:
743:
683:
618:
592:
572:
531:
520:a singular point of
429:
309:
279:
259:
233:
213:
193:
162:
136:
108:
88:
47:
1530:Commutative algebra
1476:Intersection Theory
1367:Schubert polynomial
1340:intersection theory
1000:{\displaystyle G/P}
826:{\displaystyle G/P}
1515:Algebraic geometry
1415:"Schubert variety"
1300:algebraic topology
1294:of his celebrated
1229:
1200:
1180:
1116:
1086:
1066:
1033:
997:
969:
942:
916:
896:
873:
847:
823:
792:
772:
749:
708:
669:
604:
578:
558:
446:
404:
292:
265:
245:
219:
199:
168:
148:
114:
94:
74:
29:algebraic geometry
1489:978-0-387-98549-7
1352:Schubert calculus
1278:and continued by
1203:{\displaystyle k}
1089:{\displaystyle k}
919:{\displaystyle B}
899:{\displaystyle W}
850:{\displaystyle B}
804:homogeneous space
795:{\displaystyle P}
775:{\displaystyle B}
752:{\displaystyle G}
581:{\displaystyle k}
506:algebraic variety
323:
317:
268:{\displaystyle V}
222:{\displaystyle 2}
202:{\displaystyle X}
171:{\displaystyle V}
117:{\displaystyle V}
97:{\displaystyle k}
16:(Redirected from
1542:
1501:
1467:
1427:
1409:
1276:Hermann Schubert
1238:
1236:
1235:
1230:
1228:
1227:
1222:
1209:
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1206:
1201:
1189:
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1125:
1123:
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1117:
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917:
905:
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848:
832:
830:
829:
824:
819:
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799:
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781:
779:
778:
773:
758:
756:
755:
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717:
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714:
709:
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678:
676:
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670:
662:
661:
643:
642:
630:
629:
613:
611:
610:
605:
587:
585:
584:
579:
567:
565:
564:
559:
548:
547:
542:
512:is equal to the
508:. However, when
455:
453:
452:
447:
436:
413:
411:
410:
405:
388:
387:
321:
315:
301:
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177:
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123:
121:
120:
115:
103:
101:
100:
95:
83:
81:
80:
75:
64:
63:
58:
33:Schubert variety
21:
1550:
1549:
1545:
1544:
1543:
1541:
1540:
1539:
1505:
1504:
1490:
1480:Springer-Verlag
1472:Fulton, William
1470:
1464:
1442:Fulton, William
1440:
1412:
1406:
1380:Griffiths, P.A.
1378:
1375:
1348:
1312:degeneracy loci
1272:Schubert cycles
1245:
1217:
1212:
1211:
1192:
1191:
1165:
1147:
1128:
1127:
1106:
1098:
1097:
1078:
1077:
1056:
1045:
1044:
1023:
1009:
1008:
981:
980:
959:
954:
953:
928:
927:
908:
907:
888:
887:
859:
858:
839:
838:
807:
806:
784:
783:
764:
763:
741:
740:
738:algebraic group
692:
681:
680:
653:
634:
621:
616:
615:
590:
589:
570:
569:
534:
529:
528:
427:
426:
379:
307:
306:
282:
277:
276:
257:
256:
231:
230:
211:
210:
191:
190:
184:complex numbers
160:
159:
134:
133:
126:singular points
124:, usually with
106:
105:
86:
85:
50:
45:
44:
23:
22:
15:
12:
11:
5:
1548:
1546:
1538:
1537:
1532:
1527:
1522:
1517:
1507:
1506:
1503:
1502:
1488:
1468:
1462:
1438:
1428:
1410:
1404:
1374:
1371:
1370:
1369:
1364:
1359:
1354:
1347:
1344:
1336:SchĂĽtzenberger
1308:William Fulton
1244:
1241:
1226:
1221:
1199:
1179:
1174:
1169:
1164:
1159:
1154:
1151:
1146:
1143:
1139:
1135:
1113:
1109:
1105:
1085:
1063:
1059:
1055:
1052:
1030:
1026:
1022:
1019:
1016:
996:
992:
988:
966:
962:
941:
938:
935:
915:
895:
872:
869:
866:
846:
822:
818:
814:
791:
771:
761:Borel subgroup
748:
726:-axis and the
707:
704:
699:
695:
691:
688:
668:
665:
660:
656:
652:
649:
646:
641:
637:
633:
628:
624:
603:
600:
597:
577:
557:
554:
551:
546:
541:
538:
445:
442:
439:
435:
415:
414:
403:
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386:
382:
378:
375:
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369:
366:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
320:
314:
302:nontrivially.
289:
285:
264:
244:
241:
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218:
198:
167:
147:
144:
141:
113:
93:
73:
70:
67:
62:
57:
54:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1547:
1536:
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1531:
1528:
1526:
1523:
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1518:
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1513:
1512:
1510:
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1477:
1473:
1469:
1465:
1463:9780521567244
1459:
1455:
1451:
1447:
1443:
1439:
1436:
1432:
1429:
1426:
1422:
1421:
1416:
1411:
1407:
1405:0-471-05059-8
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1376:
1372:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1349:
1345:
1343:
1341:
1337:
1333:
1329:
1326:–Gelfand and
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1288:David Hilbert
1285:
1281:
1277:
1273:
1269:
1265:
1260:
1258:
1254:
1250:
1242:
1240:
1224:
1197:
1172:
1157:
1144:
1141:
1137:
1133:
1111:
1107:
1103:
1083:
1061:
1057:
1053:
1050:
1028:
1024:
1020:
1017:
1014:
994:
990:
986:
964:
960:
939:
936:
933:
913:
893:
886:
870:
867:
864:
844:
836:
820:
816:
812:
805:
789:
769:
762:
746:
739:
736:
731:
729:
725:
721:
705:
702:
697:
693:
689:
686:
666:
663:
658:
654:
650:
647:
644:
639:
635:
631:
626:
622:
601:
598:
595:
575:
552:
544:
525:
523:
519:
515:
511:
507:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
440:
424:
420:
401:
395:
392:
384:
380:
376:
373:
367:
364:
360:
357:
354:
348:
342:
339:
336:
333:
330:
327:
318:
312:
305:
304:
303:
287:
283:
262:
242:
239:
236:
216:
196:
187:
185:
181:
165:
145:
142:
139:
131:
127:
111:
91:
68:
60:
42:
38:
35:is a certain
34:
30:
19:
18:Schubert cell
1475:
1445:
1434:
1418:
1387:
1384:Harris, J.E.
1271:
1261:
1246:
1243:Significance
835:flag variety
732:
727:
723:
719:
526:
521:
517:
513:
509:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
457:
422:
416:
188:
130:moduli space
41:Grassmannian
32:
26:
1431:H. Schubert
1296:23 problems
1210:-planes in
952:is denoted
419:real number
1509:Categories
1373:References
1126:, so that
885:Weyl group
735:semisimple
460:° ⊂
37:subvariety
1425:EMS Press
1320:Bernstein
1292:fifteenth
937:∈
868:∈
730:-plane.)
722:with the
690:
651:⊂
648:⋯
645:⊂
632:⊂
599:⊂
417:Over the
393:≥
377:∩
368:
343:
337:∣
331:⊂
240:⊂
143:⊂
1474:(1998).
1444:(1997).
1386:(1994).
1346:See also
1328:Demazure
679:, where
1498:1644323
1332:Lascoux
1324:Gelfand
1310:on the
1280:Zeuthen
1043:, with
883:of the
759:with a
1496:
1486:
1460:
1402:
1076:, the
322:
316:
180:field
39:of a
1484:ISBN
1458:ISBN
1400:ISBN
1334:and
1314:and
1302:and
31:, a
1450:doi
1392:doi
687:dim
423:xyz
365:dim
340:dim
209:of
186:.
84:of
27:In
1511::
1494:MR
1492:.
1482:.
1456:.
1433:,
1423:,
1417:,
1398:.
1382:;
1259:.
1239:.
728:xy
524:.
43:,
1500:.
1466:.
1452::
1408:.
1394::
1322:–
1225:n
1220:C
1198:k
1178:)
1173:n
1168:C
1163:(
1158:k
1153:r
1150:G
1145:=
1142:P
1138:/
1134:G
1112:n
1108:L
1104:S
1084:k
1062:k
1058:P
1054:=
1051:P
1029:n
1025:L
1021:S
1018:=
1015:G
995:P
991:/
987:G
965:w
961:X
940:W
934:w
914:B
894:W
871:W
865:w
845:B
821:P
817:/
813:G
790:P
770:B
747:G
724:x
720:L
706:j
703:=
698:j
694:V
667:V
664:=
659:n
655:V
640:2
636:V
627:1
623:V
602:V
596:w
576:k
556:)
553:V
550:(
545:k
540:r
537:G
522:X
518:L
514:x
510:L
502:X
498:x
494:L
490:X
486:x
482:L
478:X
474:L
470:x
466:L
462:X
458:X
444:)
441:V
438:(
434:P
402:.
399:}
396:1
390:)
385:2
381:V
374:w
371:(
361:,
358:2
355:=
352:)
349:w
346:(
334:V
328:w
325:{
319:=
313:X
288:2
284:V
263:V
243:V
237:w
217:2
197:X
166:V
146:V
140:w
112:V
92:k
72:)
69:V
66:(
61:k
56:r
53:G
20:)
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