401:
666:
1177:
555:
71:
1226:
443:
705:
601:
242:
145:
1030:
1063:
1259:. (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
1113:
939:
870:
966:
289:
994:
820:
1197:
1083:
913:
893:
844:
789:
769:
746:
575:
262:
216:
196:
165:
111:
91:
297:
414:-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of
606:
1508:
1476:
1118:
1450:
1392:
1523:
1256:
121:, whose elements satisfy conditions giving lower bounds to the dimensions of the intersections of its elements
1513:
1460:
1430:
1296:
1280:
1241:
1413:
1324:
1518:
1350:
505:-axis, it can be rotated or tilted around any point on the axis, and this excess of possible motions makes
519:
35:
1503:
1408:
1437:. London Mathematical Society Student Texts. Vol. 35. Cambridge, U.K.: Cambridge University Press.
723:
1403:
1202:
417:
1268:
114:
671:
1252:
1245:
580:
221:
124:
999:
1292:
1035:
1088:
918:
849:
1486:
1345:
1272:
944:
267:
1435:
Young
Tableaux. With Applications to Representation Theory and Geometry, Chapts. 5 and 9.4
8:
1355:
1328:
1304:
971:
797:
168:
1288:
1182:
1068:
898:
878:
829:
774:
754:
731:
560:
247:
201:
181:
150:
96:
76:
17:
707:. (In the example above, this would mean requiring certain intersections of the line
1472:
1446:
1388:
1368:
1340:
792:
494:
25:
1438:
1419:
1380:
1308:
1284:
1264:
1251:
The algebras of regular functions on
Schubert varieties have deep significance in
1482:
1468:
1300:
726:
1287:. The study continued in the 20th century as part of the general development of
1312:
749:
172:
1236:
Schubert varieties form one of the most important and best studied classes of
1497:
1442:
1320:
1276:
1237:
396:{\displaystyle X\ =\ \{w\subset V\mid \dim(w)=2,\,\dim(w\cap V_{2})\geq 1\}.}
1263:. The study of the intersection theory on the Grassmannian was initiated by
1424:
823:
118:
29:
1384:
1372:
407:
1240:. A certain measure of singularity of Schubert varieties is provided by
873:
557:
is defined by specifying the minimal dimension of intersection of a
1316:
661:{\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V}
1327:
in combinatorics in the 1980s, and Fulton and MacPherson in
171:, but most commonly this taken to be either the real or the
1426:
Mitt. Math. Gesellschaft
Hamburg, 1 (1889) pp. 134–155
603:
with each of the spaces in a fixed reference complete flag
264:
that intersect a fixed (reference) 2-dimensional subspace
1295:, but accelerated in the 1990s beginning with the work of
1172:{\displaystyle G/P=\mathbf {Gr} _{k}(\mathbf {C} ^{n})}
846:-orbits, which may be parametrized by certain elements
147:, with the elements of a specified complete flag. Here
1379:. Wiley Classics Library edition. Wiley-Interscience.
481:°. Since there are three degrees of freedom in moving
1205:
1185:
1121:
1091:
1071:
1038:
1002:
974:
947:
921:
901:
881:
852:
832:
800:
777:
757:
734:
674:
609:
583:
563:
522:
420:
300:
270:
250:
224:
204:
184:
153:
127:
99:
79:
38:
1331:
of singular algebraic varieties, also in the 1980s.
457:(not necessarily through the origin) which meet the
1220:
1191:
1171:
1107:
1077:
1057:
1024:
988:
960:
933:
907:
887:
864:
838:
814:
783:
763:
740:
699:
660:
595:
569:
549:
437:
395:
283:
256:
236:
210:
190:
159:
139:
105:
85:
65:
1495:
1367:
1401:
1244:, which encode their local Goresky–MacPherson
453:. This is isomorphic to the set of all lines
1307:, following up on earlier investigations of
387:
313:
1271:in the 19th century under the heading of
473:in space (while keeping contact with the
422:
352:
117:. Like the Grassmannian, it is a kind of
93:-dimensional subspaces of a vector space
167:may be a vector space over an arbitrary
1319:in representation theory in the 1970s,
1279:important enough to be included as the
1496:
1459:
1429:
516:More generally, a Schubert variety in
410:field, this can be pictured in usual
996:. The classical case corresponds to
968:and is called a Schubert variety in
722:In even greater generality, given a
550:{\displaystyle \mathbf {Gr} _{k}(V)}
66:{\displaystyle \mathbf {Gr} _{k}(V)}
13:
771:and a standard parabolic subgroup
14:
1535:
1257:algebras with a straightening law
1085:th maximal parabolic subgroup of
477:-axis) corresponds to a curve in
1377:Principles of algebraic geometry
1221:{\displaystyle \mathbf {C} ^{n}}
1208:
1156:
1141:
1138:
915:-orbit associated to an element
528:
525:
44:
41:
1231:
489:-axis, rotating, and tilting),
438:{\displaystyle \mathbb {P} (V)}
1509:Topology of homogeneous spaces
1166:
1151:
544:
538:
432:
426:
378:
359:
340:
334:
60:
54:
1:
1361:
178:A typical example is the set
1238:singular algebraic varieties
826:, consists of finitely many
700:{\displaystyle \dim V_{j}=j}
493:is a three-dimensional real
7:
1409:Encyclopedia of Mathematics
1334:
1242:Kazhdan–Lusztig polynomials
822:, which is an example of a
469:°, and continuously moving
445:, we obtain an open subset
10:
1540:
1275:. This area was deemed by
596:{\displaystyle w\subset V}
465:corresponds to a point of
244:of a 4-dimensional space
237:{\displaystyle w\subset V}
140:{\displaystyle w\subset V}
485:(moving the point on the
1443:10.1017/CBO9780511626241
1402:A.L. Onishchik (2001) ,
1351:Bott–Samelson resolution
1025:{\displaystyle G=SL_{n}}
1524:Algebraic combinatorics
1253:algebraic combinatorics
1246:intersection cohomology
1179:is the Grassmannian of
1058:{\displaystyle P=P_{k}}
791:, it is known that the
218:-dimensional subspaces
1222:
1193:
1173:
1109:
1108:{\displaystyle SL_{n}}
1079:
1059:
1026:
990:
962:
935:
934:{\displaystyle w\in W}
909:
889:
866:
865:{\displaystyle w\in W}
840:
816:
785:
765:
742:
701:
662:
597:
577:-dimensional subspace
571:
551:
461:-axis. Each such line
439:
397:
285:
258:
238:
212:
192:
161:
141:
107:
87:
67:
1514:Representation theory
1385:10.1002/9781118032527
1293:representation theory
1223:
1194:
1174:
1110:
1080:
1060:
1027:
991:
963:
961:{\displaystyle X_{w}}
936:
910:
895:. The closure of the
890:
867:
841:
817:
786:
766:
743:
702:
663:
598:
572:
552:
440:
398:
286:
284:{\displaystyle V_{2}}
259:
239:
213:
193:
162:
142:
108:
88:
68:
1467:. Berlin, New York:
1346:Bruhat decomposition
1305:Schubert polynomials
1273:enumerative geometry
1255:and are examples of
1203:
1183:
1119:
1089:
1069:
1036:
1000:
972:
945:
919:
899:
879:
850:
830:
798:
775:
755:
732:
672:
607:
581:
561:
520:
509:a singular point of
418:
298:
268:
248:
222:
202:
182:
151:
125:
97:
77:
36:
1519:Commutative algebra
1465:Intersection Theory
1356:Schubert polynomial
1329:intersection theory
989:{\displaystyle G/P}
815:{\displaystyle G/P}
1504:Algebraic geometry
1404:"Schubert variety"
1289:algebraic topology
1283:of his celebrated
1218:
1189:
1169:
1105:
1075:
1055:
1022:
986:
958:
931:
905:
885:
862:
836:
812:
781:
761:
738:
697:
658:
593:
567:
547:
435:
393:
281:
254:
234:
208:
188:
157:
137:
103:
83:
63:
18:algebraic geometry
1478:978-0-387-98549-7
1341:Schubert calculus
1267:and continued by
1192:{\displaystyle k}
1078:{\displaystyle k}
908:{\displaystyle B}
888:{\displaystyle W}
839:{\displaystyle B}
793:homogeneous space
784:{\displaystyle P}
764:{\displaystyle B}
741:{\displaystyle G}
570:{\displaystyle k}
495:algebraic variety
312:
306:
257:{\displaystyle V}
211:{\displaystyle 2}
191:{\displaystyle X}
160:{\displaystyle V}
106:{\displaystyle V}
86:{\displaystyle k}
1531:
1490:
1456:
1416:
1398:
1265:Hermann Schubert
1227:
1225:
1224:
1219:
1217:
1216:
1211:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1165:
1164:
1159:
1150:
1149:
1144:
1129:
1114:
1112:
1111:
1106:
1104:
1103:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1054:
1053:
1031:
1029:
1028:
1023:
1021:
1020:
995:
993:
992:
987:
982:
967:
965:
964:
959:
957:
956:
940:
938:
937:
932:
914:
912:
911:
906:
894:
892:
891:
886:
871:
869:
868:
863:
845:
843:
842:
837:
821:
819:
818:
813:
808:
790:
788:
787:
782:
770:
768:
767:
762:
747:
745:
744:
739:
706:
704:
703:
698:
690:
689:
667:
665:
664:
659:
651:
650:
632:
631:
619:
618:
602:
600:
599:
594:
576:
574:
573:
568:
556:
554:
553:
548:
537:
536:
531:
501:is equal to the
497:. However, when
444:
442:
441:
436:
425:
402:
400:
399:
394:
377:
376:
310:
304:
290:
288:
287:
282:
280:
279:
263:
261:
260:
255:
243:
241:
240:
235:
217:
215:
214:
209:
197:
195:
194:
189:
166:
164:
163:
158:
146:
144:
143:
138:
112:
110:
109:
104:
92:
90:
89:
84:
72:
70:
69:
64:
53:
52:
47:
22:Schubert variety
1539:
1538:
1534:
1533:
1532:
1530:
1529:
1528:
1494:
1493:
1479:
1469:Springer-Verlag
1461:Fulton, William
1453:
1431:Fulton, William
1395:
1369:Griffiths, P.A.
1364:
1337:
1301:degeneracy loci
1261:Schubert cycles
1234:
1212:
1207:
1206:
1204:
1201:
1200:
1184:
1181:
1180:
1160:
1155:
1154:
1145:
1137:
1136:
1125:
1120:
1117:
1116:
1099:
1095:
1090:
1087:
1086:
1070:
1067:
1066:
1049:
1045:
1037:
1034:
1033:
1016:
1012:
1001:
998:
997:
978:
973:
970:
969:
952:
948:
946:
943:
942:
920:
917:
916:
900:
897:
896:
880:
877:
876:
851:
848:
847:
831:
828:
827:
804:
799:
796:
795:
776:
773:
772:
756:
753:
752:
733:
730:
729:
727:algebraic group
685:
681:
673:
670:
669:
646:
642:
627:
623:
614:
610:
608:
605:
604:
582:
579:
578:
562:
559:
558:
532:
524:
523:
521:
518:
517:
421:
419:
416:
415:
372:
368:
299:
296:
295:
275:
271:
269:
266:
265:
249:
246:
245:
223:
220:
219:
203:
200:
199:
183:
180:
179:
173:complex numbers
152:
149:
148:
126:
123:
122:
115:singular points
113:, usually with
98:
95:
94:
78:
75:
74:
48:
40:
39:
37:
34:
33:
12:
11:
5:
1537:
1527:
1526:
1521:
1516:
1511:
1506:
1492:
1491:
1477:
1457:
1451:
1427:
1417:
1399:
1393:
1363:
1360:
1359:
1358:
1353:
1348:
1343:
1336:
1333:
1325:SchĂĽtzenberger
1297:William Fulton
1233:
1230:
1215:
1210:
1188:
1168:
1163:
1158:
1153:
1148:
1143:
1140:
1135:
1132:
1128:
1124:
1102:
1098:
1094:
1074:
1052:
1048:
1044:
1041:
1019:
1015:
1011:
1008:
1005:
985:
981:
977:
955:
951:
930:
927:
924:
904:
884:
861:
858:
855:
835:
811:
807:
803:
780:
760:
750:Borel subgroup
737:
715:-axis and the
696:
693:
688:
684:
680:
677:
657:
654:
649:
645:
641:
638:
635:
630:
626:
622:
617:
613:
592:
589:
586:
566:
546:
543:
540:
535:
530:
527:
434:
431:
428:
424:
404:
403:
392:
389:
386:
383:
380:
375:
371:
367:
364:
361:
358:
355:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
309:
303:
291:nontrivially.
278:
274:
253:
233:
230:
227:
207:
187:
156:
136:
133:
130:
102:
82:
62:
59:
56:
51:
46:
43:
9:
6:
4:
3:
2:
1536:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1501:
1499:
1488:
1484:
1480:
1474:
1470:
1466:
1462:
1458:
1454:
1452:9780521567244
1448:
1444:
1440:
1436:
1432:
1428:
1425:
1421:
1418:
1415:
1411:
1410:
1405:
1400:
1396:
1394:0-471-05059-8
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1365:
1357:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1338:
1332:
1330:
1326:
1322:
1318:
1315:–Gelfand and
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1277:David Hilbert
1274:
1270:
1266:
1262:
1258:
1254:
1249:
1247:
1243:
1239:
1229:
1213:
1186:
1161:
1146:
1133:
1130:
1126:
1122:
1100:
1096:
1092:
1072:
1050:
1046:
1042:
1039:
1017:
1013:
1009:
1006:
1003:
983:
979:
975:
953:
949:
928:
925:
922:
902:
882:
875:
859:
856:
853:
833:
825:
809:
805:
801:
794:
778:
758:
751:
735:
728:
725:
720:
718:
714:
710:
694:
691:
686:
682:
678:
675:
655:
652:
647:
643:
639:
636:
633:
628:
624:
620:
615:
611:
590:
587:
584:
564:
541:
533:
514:
512:
508:
504:
500:
496:
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
429:
413:
409:
390:
384:
381:
373:
369:
365:
362:
356:
353:
349:
346:
343:
337:
331:
328:
325:
322:
319:
316:
307:
301:
294:
293:
292:
276:
272:
251:
231:
228:
225:
205:
185:
176:
174:
170:
154:
134:
131:
128:
120:
116:
100:
80:
57:
49:
31:
27:
24:is a certain
23:
19:
1464:
1434:
1423:
1407:
1376:
1373:Harris, J.E.
1260:
1250:
1235:
1232:Significance
824:flag variety
721:
716:
712:
708:
515:
510:
506:
502:
498:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
411:
405:
177:
119:moduli space
30:Grassmannian
21:
15:
1420:H. Schubert
1285:23 problems
1199:-planes in
941:is denoted
408:real number
1498:Categories
1362:References
1115:, so that
874:Weyl group
724:semisimple
449:° ⊂
26:subvariety
1414:EMS Press
1309:Bernstein
1281:fifteenth
926:∈
857:∈
719:-plane.)
711:with the
679:
640:⊂
637:⋯
634:⊂
621:⊂
588:⊂
406:Over the
382:≥
366:∩
357:
332:
326:∣
320:⊂
229:⊂
132:⊂
1463:(1998).
1433:(1997).
1375:(1994).
1335:See also
1317:Demazure
668:, where
1487:1644323
1321:Lascoux
1313:Gelfand
1299:on the
1269:Zeuthen
1032:, with
872:of the
748:with a
1485:
1475:
1449:
1391:
1065:, the
311:
305:
169:field
28:of a
1473:ISBN
1447:ISBN
1389:ISBN
1323:and
1303:and
1291:and
20:, a
1439:doi
1381:doi
676:dim
412:xyz
354:dim
329:dim
198:of
175:.
73:of
16:In
1500::
1483:MR
1481:.
1471:.
1445:.
1422:,
1412:,
1406:,
1387:.
1371:;
1248:.
1228:.
717:xy
513:.
32:,
1489:.
1455:.
1441::
1397:.
1383::
1311:–
1214:n
1209:C
1187:k
1167:)
1162:n
1157:C
1152:(
1147:k
1142:r
1139:G
1134:=
1131:P
1127:/
1123:G
1101:n
1097:L
1093:S
1073:k
1051:k
1047:P
1043:=
1040:P
1018:n
1014:L
1010:S
1007:=
1004:G
984:P
980:/
976:G
954:w
950:X
929:W
923:w
903:B
883:W
860:W
854:w
834:B
810:P
806:/
802:G
779:P
759:B
736:G
713:x
709:L
695:j
692:=
687:j
683:V
656:V
653:=
648:n
644:V
629:2
625:V
616:1
612:V
591:V
585:w
565:k
545:)
542:V
539:(
534:k
529:r
526:G
511:X
507:L
503:x
499:L
491:X
487:x
483:L
479:X
475:x
471:L
467:X
463:L
459:x
455:L
451:X
447:X
433:)
430:V
427:(
423:P
391:.
388:}
385:1
379:)
374:2
370:V
363:w
360:(
350:,
347:2
344:=
341:)
338:w
335:(
323:V
317:w
314:{
308:=
302:X
277:2
273:V
252:V
232:V
226:w
206:2
186:X
155:V
135:V
129:w
101:V
81:k
61:)
58:V
55:(
50:k
45:r
42:G
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