2592:
2269:
2061:
363:
1847:). The kernel of the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements in the image of the canonical map
474:
1244:
1157:
864:
2382:
1728:
547:
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
111:. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags.
709:. Any complete flag then splits into a direct sum of one-dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the
2100:
1075:
is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space
2363:
2341:
2120:
757:
1912:
1265:), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it.
275:
2766:, this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano.
2962:
686:) of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a
392:
2102:
denotes positive-degree elements and the parentheses the generation of an ideal. For example, for the complete complex flag manifold
2796:
Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are
Riemannian symmetric spaces under
1303:. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other.
107:. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the
2293:
1168:
1081:
91:
Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space
796:
2957:
1644:
1380:
come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space.
266:
2587:{\displaystyle H^{*}{\big (}U(2)/T^{2}{\big )}\cong \mathbb {Q} /(t_{1}+t_{2},t_{1}t_{2})\cong \mathbb {Q} /(t_{1}^{2}),}
2630:
1432:
1653:
575:
484:
2845:
2800:
are known as symmetric R-spaces. The symmetric R-spaces which are not
Hermitian symmetric are obtained by taking
2778:
618:
228:
1368:
is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the
2857:
205:
146:
674:
Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the
1538:
1339:
626:
1431:
is another compact real manifold. (Both are actually complex homogeneous spaces in a canonical way through
2069:
1526:
1498:
1369:
1276:
582:
235:-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric
2869:
2685:
acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety
248:
44:
2633:. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for
2874:
1601:
675:
572:
185:
100:
2264:{\displaystyle H^{*}{\big (}U(n)/T^{n}{\big )}\cong \mathbb {Q} /(\sigma _{1},\ldots ,\sigma _{n}),}
1356:
in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(
2694:
1319:
622:
201:
55:
2346:
2324:
2920:
2372:. Then we expect the cohomology ring to be an exterior algebra on a generator of degree two (the
2056:{\displaystyle H^{*}(G/H)\cong H^{*}(BT)^{W(H)}/{\big (}{\widetilde {H}}^{*}(BT)^{W(G)}{\big )},}
1733:
of the spectral sequence must eventually take the space of primitive elements in the left column
774:-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(
85:
2725:
718:
1364:). For orthogonal groups there is a similar picture, with a couple of complications. First, if
2373:
1797:
1483:
1439:
1408:
710:
639:
217:
197:
131:
40:
2837:
2648:
2618:
1640:
1463:
1327:
1323:
1273:
108:
70:
1454:
is concentrated in even degrees, but in fact, something much stronger can be said. Because
2763:
2668:
2643:
showed that this characterizes the flag varieties of a general semisimple algebraic group
1765:
1443:
262:
224:
213:
135:
66:
2913:
687:
609:
invertible matrices. The standard flag associated with this basis is the one where the
556:
181:
170:
158:
2951:
1393:
767:
698:
571:
are no different from each other from a geometric point of view. That is to say, the
193:
119:
358:{\displaystyle \{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.}
2640:
2318:
1628:
1542:
1003:
935:
48:
2925:
2904:
1306:
The extension of the terminology "flag variety" is reasonable, because points of
782:), and T by the diagonal orthogonal matrices (which have diagonal entries ±1).
706:
28:
1056:), and thus the partial flag variety is isomorphic to the homogeneous space SL(
2895:
1903:
1801:
1827:) is trivial in this case, and the characteristic map is surjective, so that
2805:
1348:
1280:
1257:
The upper triangular matrices of determinant one are a Borel subgroup of SL(
17:
114:
In the most general sense, a generalized flag variety is defined to mean a
180:. A projective homogeneous variety may also be realised as the orbit of a
1044:
Restricting to matrices of determinant one, this is a parabolic subgroup
1772:
full rank on the primitive subspace, then the image of the bottom row
265:, where "increasing" means each is a proper subspace of the next (see
2849:
227:. Over the complex numbers, the corresponding flag manifolds are the
1006:
lower triangular matrices, where the dimensions of the blocks are
2888:
The
Penrose Transform: its Interaction with Representation Theory
2655:, or equivalently (in this context), the projective homogeneous
469:{\displaystyle 0=d_{0}<d_{1}<d_{2}<\cdots <d_{k}=n,}
65:
is the real or complex numbers, a generalized flag variety is a
2754:
admit a strictly larger Lie group of transformations, namely
141:(and smooth stabilizer subgroup; that is no restriction for
2929:, Graduate Texts in Mathematics, 21, Springer-Verlag, 1972.
2613:
is a (finite dimensional) highest weight representation of
1800:
again, because a compact homogeneous space admits a finite
2601:
Highest weight orbits and projective homogeneous varieties
1239:{\displaystyle O(n)/O(n_{1})\times \cdots \times O(n_{k})}
1152:{\displaystyle U(n)/U(n_{1})\times \cdots \times U(n_{k})}
638:. The complete flag variety can therefore be written as a
952:
This is a homogeneous space for the general linear group
2677:
be a semisimple Lie group with maximal compact subgroup
859:{\displaystyle F(d_{1},d_{2},\ldots d_{k},\mathbb {F} )}
2731:
Turning this around, the
Riemannian homogeneous spaces
1623:
Let us now restrict our coefficient ring to be a field
1352:
if the symplectic form vanishes on proper subspaces of
192:. The complex projective homogeneous varieties are the
2385:
2349:
2327:
2123:
2072:
1915:
1792:) of the sequence would be infinite-dimensional as a
1656:
1171:
1084:
913:. The complete flag variety is the special case that
799:
721:
395:
278:
1796:-vector space, which is impossible, for instance by
2609:is a semisimple algebraic group (or Lie group) and
1541:of this bundle to understand the fiber-restriction
697:is the real or complex numbers we can introduce an
2617:, then the highest weight space is a point in the
2586:
2357:
2335:
2263:
2094:
2055:
1768:, so if the collection of edge homomorphisms were
1722:
1238:
1151:
858:
751:
659:, which shows in particular that it has dimension
617:vectors of the basis. Relative to this basis, the
468:
357:
2934:On filtered Lie algebras and geometric structures
1906:, so one finally obtains the concise description
1392:is a compact, connected Lie group, it contains a
2902:Jürgen Berndt, Sergio Console and Carlos Olmos,
1902:) of elements invariant under the action of the
1723:{\displaystyle E_{r+1}^{0,r}\to E_{r+1}^{r+1,0}}
984:). The stabilizer of a flag of nested subspaces
1372:. Second, for vector spaces of even dimension 2
578:transitively on the set of all complete flags.
2773:is a complex Lie group, the symmetric spaces
2432:
2398:
2170:
2136:
2045:
1991:
1643:on generators of odd degree (the subspace of
593:, whose general linear group is the group GL(
8:
1002:can be taken to be the group of nonsingular
285:
279:
253:A flag in a finite dimensional vector space
1647:). It follows that the edge homomorphisms
1415:is any other closed, connected subgroup of
2914:Lectures on the geometry of flag varieties
2886:Robert J. Baston and Michael G. Eastwood,
1283:, then the (generalized) flag variety for
2572:
2567:
2555:
2546:
2535:
2534:
2522:
2512:
2499:
2486:
2474:
2465:
2452:
2441:
2440:
2431:
2430:
2424:
2415:
2397:
2396:
2390:
2384:
2351:
2350:
2348:
2329:
2328:
2326:
2249:
2230:
2218:
2209:
2190:
2179:
2178:
2169:
2168:
2162:
2153:
2135:
2134:
2128:
2122:
2086:
2075:
2074:
2071:
2044:
2043:
2028:
2009:
1998:
1997:
1990:
1989:
1984:
1969:
1950:
1932:
1920:
1914:
1702:
1691:
1672:
1661:
1655:
1314:can still be described using flags. When
1227:
1199:
1184:
1170:
1140:
1112:
1097:
1083:
849:
848:
839:
823:
810:
798:
743:
734:
720:
451:
432:
419:
406:
394:
340:
321:
308:
295:
277:
1438:The presence of a complex structure and
1330:, this is particularly transparent. If (
869:is the space of all flags of signature (
200:of parabolic type. They are homogeneous
1529:of the right multiplication action of
2908:, Chapman & Hall/CRC Press, 2003.
778:) replaced by the orthogonal group O(
7:
2777:arising in this way are the compact
2625:) and its orbit under the action of
2305:. For a more concrete example, take
2095:{\displaystyle {\widetilde {H}}^{*}}
1627:of characteristic zero, so that, by
1584:), so called because its image, the
613:th subspace is spanned by the first
551:Prototype: the complete flag variety
1537:, and we can use the cohomological
1376:, isotropic subspaces of dimension
1253:Generalization to semisimple groups
99:, which is a flag variety for the
25:
1890:) is injective, and likewise for
1470:, there exists a classifying map
2917:, Lecture notes, Varsovie, 2003.
2890:, Oxford University Press, 1989.
2758:. Specializing to the case that
2294:elementary symmetric polynomials
1748:bijectively into the bottom row
2812:of a Hermitian symmetric space
2789:is the biholomorphism group of
1411:is a compact real manifold. If
559:, any two complete flags in an
84:. Flag varieties are naturally
2896:Lie group actions on manifolds
2578:
2560:
2552:
2539:
2528:
2479:
2471:
2445:
2412:
2406:
2255:
2223:
2215:
2183:
2150:
2144:
2038:
2032:
2025:
2015:
1979:
1973:
1966:
1956:
1940:
1926:
1863:) induced by the inclusion of
1684:
1233:
1220:
1205:
1192:
1181:
1175:
1146:
1133:
1118:
1105:
1094:
1088:
853:
803:
731:
725:
705:such that the chosen basis is
555:According to basic results of
116:projective homogeneous variety
1:
2899:, Lecture notes, Tokyo, 2002.
1564:) and the characteristic map
1442:make it easy to see that the
528:of the flag is the sequence (
261:is an increasing sequence of
212:, and they are precisely the
2963:Algebraic homogeneous spaces
2932:S. Kobayashi and T. Nagano,
2808:of the biholomorphism group
2631:projective algebraic variety
2358:{\displaystyle \mathbb {C} }
2336:{\displaystyle \mathbb {C} }
621:of the standard flag is the
231:. Over the real numbers, an
2832:is a parabolic subgroup of
2785:is the isometry group, and
1299:is a parabolic subgroup of
520:, otherwise it is called a
161:, then it is isomorphic to
2979:
2846:projective transformations
2779:Hermitian symmetric spaces
2666:
1268:Hence, more generally, if
945:-dimensional subspaces of
752:{\displaystyle U(n)/T^{n}}
563:-dimensional vector space
246:
229:Hermitian symmetric spaces
184:vector in a projectivized
2905:Submanifolds and Holonomy
2858:conformal transformations
2693:is a compact homogeneous
2647:: they are precisely the
2283:are of degree 2 and the σ
1894:, with image the subring
790:The partial flag variety
627:lower triangular matrices
2716:is a complex Lie group,
1521:, we obtain a principal
1407:of left cosets with the
1162:in the complex case, or
206:maximal compact subgroup
47:in a finite-dimensional
33:generalized flag variety
2926:Linear Algebraic Groups
1604:of the original bundle
1539:Serre spectral sequence
1370:split orthogonal groups
1342:then a partial flag in
1340:symplectic vector space
964:. To be explicit, take
243:Flags in a vector space
2936:I, II, J. Math. Mech.
2708:) with isometry group
2651:homogeneous spaces of
2588:
2359:
2337:
2265:
2096:
2057:
1724:
1602:characteristic classes
1586:characteristic subring
1240:
1153:
860:
786:Partial flag varieties
753:
589:, identifying it with
491:. Hence, we must have
470:
359:
223:Flag manifolds can be
196:flat model spaces for
2958:Differential geometry
2870:Parabolic Lie algebra
2589:
2360:
2338:
2266:
2097:
2058:
1725:
1440:cellular (co)homology
1241:
1154:
861:
754:
629:, which we denote by
499:. A flag is called a
471:
360:
249:flag (linear algebra)
2875:Bruhat decomposition
2836:. Examples include
2637:arises in this way.
2383:
2347:
2325:
2121:
2070:
1913:
1780:) in the final page
1654:
1169:
1082:
892:) in a vector space
797:
719:
676:special linear group
573:general linear group
393:
368:If we write the dim
276:
202:Riemannian manifolds
101:special linear group
86:projective varieties
2695:Riemannian manifold
2577:
1839:) is a quotient of
1798:cellular cohomology
1719:
1683:
1525:-bundle called the
122:projective variety
2921:James E. Humphreys
2712:. Furthermore, if
2584:
2563:
2355:
2333:
2317:)/ is the complex
2261:
2092:
2053:
1807:Thus the ring map
1720:
1687:
1657:
1645:primitive elements
1249:in the real case.
1236:
1149:
856:
749:
466:
355:
218:compact Lie groups
171:parabolic subgroup
2940:(1964), 875–907,
2838:projective spaces
2724:is a homogeneous
2374:fundamental class
2296:in the variables
2083:
2006:
1499:homotopy quotient
1484:classifying space
1409:quotient topology
711:homogeneous space
667:−1)/2 over
640:homogeneous space
198:Cartan geometries
132:transitive action
43:whose points are
41:homogeneous space
16:(Redirected from
2970:
2663:Symmetric spaces
2619:projective space
2593:
2591:
2590:
2585:
2576:
2571:
2559:
2551:
2550:
2538:
2527:
2526:
2517:
2516:
2504:
2503:
2491:
2490:
2478:
2470:
2469:
2457:
2456:
2444:
2436:
2435:
2429:
2428:
2419:
2402:
2401:
2395:
2394:
2364:
2362:
2361:
2356:
2354:
2342:
2340:
2339:
2334:
2332:
2270:
2268:
2267:
2262:
2254:
2253:
2235:
2234:
2222:
2214:
2213:
2195:
2194:
2182:
2174:
2173:
2167:
2166:
2157:
2140:
2139:
2133:
2132:
2101:
2099:
2098:
2093:
2091:
2090:
2085:
2084:
2076:
2062:
2060:
2059:
2054:
2049:
2048:
2042:
2041:
2014:
2013:
2008:
2007:
1999:
1995:
1994:
1988:
1983:
1982:
1955:
1954:
1936:
1925:
1924:
1729:
1727:
1726:
1721:
1718:
1701:
1682:
1671:
1641:exterior algebra
1509:in the sequence
1489:. If we replace
1482:with target the
1433:complexification
1328:orthogonal group
1324:symplectic group
1245:
1243:
1242:
1237:
1232:
1231:
1204:
1203:
1188:
1158:
1156:
1155:
1150:
1145:
1144:
1117:
1116:
1101:
865:
863:
862:
857:
852:
844:
843:
828:
827:
815:
814:
758:
756:
755:
750:
748:
747:
738:
475:
473:
472:
467:
456:
455:
437:
436:
424:
423:
411:
410:
364:
362:
361:
356:
345:
344:
326:
325:
313:
312:
300:
299:
225:symmetric spaces
214:coadjoint orbits
109:symplectic group
71:complex manifold
21:
2978:
2977:
2973:
2972:
2971:
2969:
2968:
2967:
2948:
2947:
2944:(1965) 513–521.
2893:Jürgen Berndt,
2883:
2866:
2764:symmetric space
2726:Kähler manifold
2671:
2669:Symmetric space
2665:
2603:
2542:
2518:
2508:
2495:
2482:
2461:
2448:
2420:
2386:
2381:
2380:
2376:), and indeed,
2345:
2344:
2323:
2322:
2304:
2288:
2282:
2245:
2226:
2205:
2186:
2158:
2124:
2119:
2118:
2073:
2068:
2067:
2024:
1996:
1965:
1946:
1916:
1911:
1910:
1747:
1652:
1651:
1600:), carries the
1527:Borel fibration
1508:
1444:cohomology ring
1386:
1320:classical group
1255:
1223:
1195:
1167:
1166:
1136:
1108:
1080:
1079:
1040:
1033:
1023:
1014:
1001:
992:
944:
921:
908:
891:
882:
875:
835:
819:
806:
795:
794:
788:
739:
717:
716:
658:
637:
625:of nonsingular
581:Fix an ordered
553:
543:
534:
511:
447:
428:
415:
402:
391:
390:
385:
376:
336:
317:
304:
291:
274:
273:
251:
245:
136:reductive group
23:
22:
15:
12:
11:
5:
2976:
2974:
2966:
2965:
2960:
2950:
2949:
2946:
2945:
2930:
2918:
2911:Michel Brion,
2909:
2900:
2891:
2882:
2879:
2878:
2877:
2872:
2865:
2862:
2752:
2751:
2667:Main article:
2664:
2661:
2602:
2599:
2595:
2594:
2583:
2580:
2575:
2570:
2566:
2562:
2558:
2554:
2549:
2545:
2541:
2537:
2533:
2530:
2525:
2521:
2515:
2511:
2507:
2502:
2498:
2494:
2489:
2485:
2481:
2477:
2473:
2468:
2464:
2460:
2455:
2451:
2447:
2443:
2439:
2434:
2427:
2423:
2418:
2414:
2411:
2408:
2405:
2400:
2393:
2389:
2353:
2331:
2300:
2289:are the first
2284:
2278:
2272:
2271:
2260:
2257:
2252:
2248:
2244:
2241:
2238:
2233:
2229:
2225:
2221:
2217:
2212:
2208:
2204:
2201:
2198:
2193:
2189:
2185:
2181:
2177:
2172:
2165:
2161:
2156:
2152:
2149:
2146:
2143:
2138:
2131:
2127:
2089:
2082:
2079:
2064:
2063:
2052:
2047:
2040:
2037:
2034:
2031:
2027:
2023:
2020:
2017:
2012:
2005:
2002:
1993:
1987:
1981:
1978:
1975:
1972:
1968:
1964:
1961:
1958:
1953:
1949:
1945:
1942:
1939:
1935:
1931:
1928:
1923:
1919:
1764:have the same
1745:
1741:) of the page
1731:
1730:
1717:
1714:
1711:
1708:
1705:
1700:
1697:
1694:
1690:
1686:
1681:
1678:
1675:
1670:
1667:
1664:
1660:
1629:Hopf's theorem
1504:
1399:and the space
1385:
1382:
1254:
1251:
1247:
1246:
1235:
1230:
1226:
1222:
1219:
1216:
1213:
1210:
1207:
1202:
1198:
1194:
1191:
1187:
1183:
1180:
1177:
1174:
1160:
1159:
1148:
1143:
1139:
1135:
1132:
1129:
1126:
1123:
1120:
1115:
1111:
1107:
1104:
1100:
1096:
1093:
1090:
1087:
1038:
1028:
1019:
1010:
997:
988:
942:
934:=2, this is a
917:
904:
887:
880:
873:
867:
866:
855:
851:
847:
842:
838:
834:
831:
826:
822:
818:
813:
809:
805:
802:
787:
784:
760:
759:
746:
742:
737:
733:
730:
727:
724:
688:Borel subgroup
654:
633:
557:linear algebra
552:
549:
539:
532:
507:
477:
476:
465:
462:
459:
454:
450:
446:
443:
440:
435:
431:
427:
422:
418:
414:
409:
405:
401:
398:
381:
372:
366:
365:
354:
351:
348:
343:
339:
335:
332:
329:
324:
320:
316:
311:
307:
303:
298:
294:
290:
287:
284:
281:
247:Main article:
244:
241:
186:representation
182:highest weight
159:rational point
147:characteristic
24:
14:
13:
10:
9:
6:
4:
3:
2:
2975:
2964:
2961:
2959:
2956:
2955:
2953:
2943:
2939:
2935:
2931:
2928:
2927:
2922:
2919:
2916:
2915:
2910:
2907:
2906:
2901:
2898:
2897:
2892:
2889:
2885:
2884:
2880:
2876:
2873:
2871:
2868:
2867:
2863:
2861:
2859:
2856:the group of
2855:
2851:
2847:
2844:the group of
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2815:
2811:
2807:
2803:
2799:
2794:
2792:
2788:
2784:
2780:
2776:
2772:
2767:
2765:
2761:
2757:
2749:
2745:
2741:
2737:
2734:
2733:
2732:
2729:
2727:
2723:
2719:
2715:
2711:
2707:
2703:
2699:
2696:
2692:
2688:
2684:
2680:
2676:
2670:
2662:
2660:
2658:
2654:
2650:
2646:
2642:
2638:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2600:
2598:
2581:
2573:
2568:
2564:
2556:
2547:
2543:
2531:
2523:
2519:
2513:
2509:
2505:
2500:
2496:
2492:
2487:
2483:
2475:
2466:
2462:
2458:
2453:
2449:
2437:
2425:
2421:
2416:
2409:
2403:
2391:
2387:
2379:
2378:
2377:
2375:
2371:
2367:
2320:
2316:
2312:
2309:= 2, so that
2308:
2303:
2299:
2295:
2292:
2287:
2281:
2277:
2258:
2250:
2246:
2242:
2239:
2236:
2231:
2227:
2219:
2210:
2206:
2202:
2199:
2196:
2191:
2187:
2175:
2163:
2159:
2154:
2147:
2141:
2129:
2125:
2117:
2116:
2115:
2113:
2109:
2105:
2087:
2080:
2077:
2050:
2035:
2029:
2021:
2018:
2010:
2003:
2000:
1985:
1976:
1970:
1962:
1959:
1951:
1947:
1943:
1937:
1933:
1929:
1921:
1917:
1909:
1908:
1907:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1872:
1870:
1866:
1862:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1805:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1744:
1740:
1736:
1715:
1712:
1709:
1706:
1703:
1698:
1695:
1692:
1688:
1679:
1676:
1673:
1668:
1665:
1662:
1658:
1650:
1649:
1648:
1646:
1642:
1638:
1634:
1630:
1626:
1621:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1507:
1503:
1500:
1496:
1492:
1488:
1485:
1481:
1477:
1473:
1469:
1467:
1461:
1457:
1453:
1449:
1445:
1441:
1436:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1395:
1394:maximal torus
1391:
1383:
1381:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1350:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1304:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1275:
1271:
1266:
1264:
1260:
1252:
1250:
1228:
1224:
1217:
1214:
1211:
1208:
1200:
1196:
1189:
1185:
1178:
1172:
1165:
1164:
1163:
1141:
1137:
1130:
1127:
1124:
1121:
1113:
1109:
1102:
1098:
1091:
1085:
1078:
1077:
1076:
1074:
1069:
1067:
1063:
1059:
1055:
1051:
1047:
1042:
1037:
1031:
1027:
1022:
1018:
1013:
1009:
1005:
1000:
996:
993:of dimension
991:
987:
983:
979:
975:
971:
967:
963:
959:
955:
950:
948:
941:
937:
933:
929:
925:
920:
916:
912:
907:
903:
899:
896:of dimension
895:
890:
886:
879:
872:
845:
840:
836:
832:
829:
824:
820:
816:
811:
807:
800:
793:
792:
791:
785:
783:
781:
777:
773:
770:and T is the
769:
768:unitary group
765:
744:
740:
735:
728:
722:
715:
714:
713:
712:
708:
704:
700:
699:inner product
696:
693:If the field
691:
689:
685:
681:
677:
672:
670:
666:
662:
657:
653:
649:
645:
641:
636:
632:
628:
624:
620:
616:
612:
608:
604:
600:
596:
592:
588:
584:
579:
577:
574:
570:
567:over a field
566:
562:
558:
550:
548:
545:
542:
538:
531:
527:
523:
519:
515:
510:
506:
502:
501:complete flag
498:
494:
490:
486:
482:
463:
460:
457:
452:
448:
444:
441:
438:
433:
429:
425:
420:
416:
412:
407:
403:
399:
396:
389:
388:
387:
386:then we have
384:
380:
375:
371:
352:
349:
346:
341:
337:
333:
330:
327:
322:
318:
314:
309:
305:
301:
296:
292:
288:
282:
272:
271:
270:
268:
264:
260:
257:over a field
256:
250:
242:
240:
238:
234:
230:
226:
221:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
172:
168:
164:
160:
156:
152:
148:
144:
140:
137:
133:
129:
126:over a field
125:
121:
118:, that is, a
117:
112:
110:
106:
102:
98:
95:over a field
94:
89:
87:
83:
82:flag manifold
80:
76:
72:
68:
64:
60:
57:
53:
50:
46:
42:
38:
34:
30:
19:
2941:
2937:
2933:
2924:
2912:
2903:
2894:
2887:
2853:
2841:
2833:
2829:
2825:
2821:
2817:
2813:
2809:
2801:
2797:
2795:
2790:
2786:
2782:
2774:
2770:
2768:
2759:
2755:
2753:
2747:
2743:
2739:
2735:
2730:
2721:
2717:
2713:
2709:
2705:
2701:
2697:
2690:
2686:
2682:
2678:
2674:
2672:
2659:-varieties.
2656:
2652:
2644:
2641:Armand Borel
2639:
2634:
2626:
2622:
2614:
2610:
2606:
2604:
2596:
2369:
2365:
2319:Grassmannian
2314:
2310:
2306:
2301:
2297:
2290:
2285:
2279:
2275:
2273:
2111:
2107:
2103:
2065:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1873:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1806:
1802:CW structure
1793:
1789:
1785:
1781:
1777:
1773:
1769:
1761:
1757:
1753:
1749:
1742:
1738:
1734:
1732:
1636:
1632:
1624:
1622:
1617:
1613:
1609:
1605:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1561:
1557:
1553:
1549:
1545:
1543:homomorphism
1534:
1530:
1522:
1518:
1514:
1510:
1505:
1501:
1494:
1490:
1486:
1479:
1475:
1471:
1465:
1459:
1455:
1451:
1447:
1437:
1428:
1424:
1420:
1416:
1412:
1404:
1400:
1396:
1389:
1387:
1377:
1373:
1365:
1361:
1357:
1353:
1347:
1343:
1335:
1331:
1322:, such as a
1315:
1311:
1307:
1305:
1300:
1296:
1292:
1288:
1284:
1269:
1267:
1262:
1258:
1256:
1248:
1161:
1072:
1070:
1065:
1061:
1057:
1053:
1049:
1045:
1043:
1035:
1029:
1025:
1020:
1016:
1011:
1007:
998:
994:
989:
985:
981:
977:
973:
969:
965:
961:
957:
953:
951:
946:
939:
936:Grassmannian
931:
927:
923:
918:
914:
910:
905:
901:
897:
893:
888:
884:
877:
870:
868:
789:
779:
775:
771:
763:
761:
702:
694:
692:
683:
679:
673:
668:
664:
660:
655:
651:
647:
643:
634:
630:
614:
610:
606:
602:
598:
594:
590:
586:
580:
568:
564:
560:
554:
546:
540:
536:
529:
525:
522:partial flag
521:
517:
513:
508:
504:
500:
496:
492:
488:
480:
478:
382:
378:
373:
369:
367:
258:
254:
252:
236:
232:
222:
209:
189:
177:
173:
166:
162:
154:
150:
142:
138:
127:
123:
115:
113:
104:
96:
92:
90:
81:
78:
74:
62:
58:
51:
49:vector space
37:flag variety
36:
32:
26:
18:Flag variety
2114:, one has
1756:): we know
1419:containing
707:orthonormal
73:, called a
35:(or simply
29:mathematics
2952:Categories
2881:References
2820:such that
2597:as hoped.
2274:where the
1904:Weyl group
1464:principal
1384:Cohomology
1274:semisimple
619:stabilizer
267:filtration
204:under any
149:zero). If
2824: :=
2806:real form
2532:≅
2438:≅
2392:∗
2247:σ
2240:…
2228:σ
2200:…
2176:≅
2130:∗
2088:∗
2081:~
2011:∗
2004:~
1952:∗
1944:≅
1922:∗
1685:→
1497:with the
1349:isotropic
1281:Lie group
1277:algebraic
1215:×
1212:⋯
1209:×
1128:×
1125:⋯
1122:×
1015: :=
833:…
766:) is the
526:signature
485:dimension
442:⋯
334:⊂
331:⋯
328:⊂
315:⊂
302:⊂
263:subspaces
239:-spaces.
169:for some
2864:See also
2804:to be a
2649:complete
1874:The map
1639:) is an
1032:−1
1024:−
972:so that
926:for all
762:where U(
516:for all
2850:spheres
2828:∩
2746:∩
2704:∩
2681:. Then
1468:-bundle
1423:, then
1338:) is a
930:. When
535:, ...,
483:is the
194:compact
153:has an
130:with a
79:complex
61:. When
54:over a
39:) is a
2852:(with
2848:) and
2840:(with
2066:where
1362:ω
1336:ω
1295:where
1048:of SL(
1041:= 0).
1034:(with
883:, ...
524:. The
479:where
120:smooth
67:smooth
2762:is a
2629:is a
2321:Gr(1,
1462:is a
1318:is a
1272:is a
1004:block
976:= GL(
960:over
909:over
623:group
601:) of
583:basis
134:of a
103:over
56:field
45:flags
2673:Let
2343:) ≈
1882:) →
1855:) →
1819:) →
1766:rank
1760:and
1572:) →
1556:) →
1435:.)
650:) /
585:for
576:acts
445:<
439:<
426:<
413:<
75:real
31:, a
2860:).
2769:If
2605:If
1871:.
1867:in
1770:not
1588:of
1533:on
1515:G/H
1460:G/H
1446:of
1388:If
1346:is
1326:or
1287:is
1279:or
1071:If
956:of
938:of
701:on
678:SL(
642:GL(
544:).
503:if
487:of
269:):
216:of
208:of
188:of
176:of
145:of
77:or
69:or
27:In
2954::
2942:14
2938:13
2923:,
2793:.
2781::
2742:/(
2738:=
2728:.
2700:/(
2621:P(
2368:≈
2110:)/
1900:BT
1898:*(
1888:BT
1886:*(
1880:BG
1878:*(
1861:BH
1859:*(
1853:BG
1851:*(
1845:BH
1843:*(
1831:*(
1823:*(
1811:*(
1804:.
1784:*(
1778:BH
1776:*(
1754:BH
1752:*(
1737:*(
1635:*(
1631:,
1620:.
1612:→
1608:→
1592:*(
1576:*(
1570:BH
1568:*(
1560:*(
1548:*(
1519:BH
1517:→
1513:→
1487:BH
1480:BH
1478:→
1458:→
1334:,
1068:.
1064:)/
968:=
949:.
922:=
900:=
876:,
690:.
671:.
605:×
512:=
495:≤
377:=
220:.
88:.
2854:G
2842:G
2834:G
2830:G
2826:P
2822:P
2818:P
2816:/
2814:G
2810:G
2802:G
2798:K
2791:M
2787:G
2783:K
2775:M
2771:G
2760:M
2756:G
2750:)
2748:P
2744:K
2740:K
2736:M
2722:P
2720:/
2718:G
2714:G
2710:K
2706:P
2702:K
2698:K
2691:P
2689:/
2687:G
2683:K
2679:K
2675:G
2657:G
2653:G
2645:G
2635:G
2627:G
2623:V
2615:G
2611:V
2607:G
2582:,
2579:)
2574:2
2569:1
2565:t
2561:(
2557:/
2553:]
2548:1
2544:t
2540:[
2536:Q
2529:)
2524:2
2520:t
2514:1
2510:t
2506:,
2501:2
2497:t
2493:+
2488:1
2484:t
2480:(
2476:/
2472:]
2467:2
2463:t
2459:,
2454:1
2450:t
2446:[
2442:Q
2433:)
2426:2
2422:T
2417:/
2413:)
2410:2
2407:(
2404:U
2399:(
2388:H
2370:S
2366:P
2352:C
2330:C
2315:2
2313:(
2311:U
2307:n
2302:j
2298:t
2291:n
2286:j
2280:j
2276:t
2259:,
2256:)
2251:n
2243:,
2237:,
2232:1
2224:(
2220:/
2216:]
2211:n
2207:t
2203:,
2197:,
2192:1
2188:t
2184:[
2180:Q
2171:)
2164:n
2160:T
2155:/
2151:)
2148:n
2145:(
2142:U
2137:(
2126:H
2112:T
2108:n
2106:(
2104:U
2078:H
2051:,
2046:)
2039:)
2036:G
2033:(
2030:W
2026:)
2022:T
2019:B
2016:(
2001:H
1992:(
1986:/
1980:)
1977:H
1974:(
1971:W
1967:)
1963:T
1960:B
1957:(
1948:H
1941:)
1938:H
1934:/
1930:G
1927:(
1918:H
1896:H
1892:H
1884:H
1876:H
1869:G
1865:H
1857:H
1849:H
1841:H
1837:H
1835:/
1833:G
1829:H
1825:G
1821:H
1817:H
1815:/
1813:G
1809:H
1794:k
1790:H
1788:/
1786:G
1782:H
1774:H
1762:H
1758:G
1750:H
1746:2
1743:E
1739:G
1735:H
1716:0
1713:,
1710:1
1707:+
1704:r
1699:1
1696:+
1693:r
1689:E
1680:r
1677:,
1674:0
1669:1
1666:+
1663:r
1659:E
1637:G
1633:H
1625:k
1618:H
1616:/
1614:G
1610:G
1606:H
1598:H
1596:/
1594:G
1590:H
1582:H
1580:/
1578:G
1574:H
1566:H
1562:G
1558:H
1554:H
1552:/
1550:G
1546:H
1535:G
1531:H
1523:G
1511:G
1506:H
1502:G
1495:H
1493:/
1491:G
1476:H
1474:/
1472:G
1466:H
1456:G
1452:H
1450:/
1448:G
1429:H
1427:/
1425:G
1421:T
1417:G
1413:H
1405:T
1403:/
1401:G
1397:T
1390:G
1378:m
1374:m
1366:F
1360:,
1358:V
1354:V
1344:V
1332:V
1316:G
1312:P
1310:/
1308:G
1301:G
1297:P
1293:P
1291:/
1289:G
1285:G
1270:G
1263:F
1261:,
1259:n
1234:)
1229:k
1225:n
1221:(
1218:O
1206:)
1201:1
1197:n
1193:(
1190:O
1186:/
1182:)
1179:n
1176:(
1173:O
1147:)
1142:k
1138:n
1134:(
1131:U
1119:)
1114:1
1110:n
1106:(
1103:U
1099:/
1095:)
1092:n
1089:(
1086:U
1073:F
1066:P
1062:F
1060:,
1058:n
1054:F
1052:,
1050:n
1046:P
1039:0
1036:d
1030:i
1026:d
1021:i
1017:d
1012:i
1008:n
999:i
995:d
990:i
986:V
982:F
980:,
978:n
974:G
970:F
966:V
962:F
958:V
954:G
947:V
943:1
940:d
932:k
928:i
924:i
919:i
915:d
911:F
906:k
902:d
898:n
894:V
889:k
885:d
881:2
878:d
874:1
871:d
854:)
850:F
846:,
841:k
837:d
830:,
825:2
821:d
817:,
812:1
808:d
804:(
801:F
780:n
776:n
772:n
764:n
745:n
741:T
736:/
732:)
729:n
726:(
723:U
703:V
695:F
684:F
682:,
680:n
669:F
665:n
663:(
661:n
656:n
652:B
648:F
646:,
644:n
635:n
631:B
615:i
611:i
607:n
603:n
599:F
597:,
595:n
591:F
587:V
569:F
565:V
561:n
541:k
537:d
533:1
530:d
518:i
514:i
509:i
505:d
497:n
493:k
489:V
481:n
464:,
461:n
458:=
453:k
449:d
434:2
430:d
421:1
417:d
408:0
404:d
400:=
397:0
383:i
379:d
374:i
370:V
353:.
350:V
347:=
342:k
338:V
323:2
319:V
310:1
306:V
297:0
293:V
289:=
286:}
283:0
280:{
259:F
255:V
237:R
233:R
210:G
190:G
178:G
174:P
167:P
165:/
163:G
157:-
155:F
151:X
143:F
139:G
128:F
124:X
105:F
97:F
93:V
63:F
59:F
52:V
20:)
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