822:
625:
2614:
2334:
1137:. The image can be completely characterized as the intersection of a number of quadrics, the Plücker quadrics (see below), which are expressed by homogeneous quadratic relations on the Plücker coordinates (see below) that derive from
817:{\displaystyle {\begin{aligned}\iota \colon \mathbf {Gr} (k,V)&{}\rightarrow \mathbf {P} ({\textstyle \bigwedge }^{k}V),\\\iota \colon {\mathcal {W}}:=\operatorname {span} (w_{1},\ldots ,w_{k})&{}\mapsto ,\end{aligned}}}
1047:
3164:
1286:
1241:
1135:
194:
3104:
1998:
2704:
2965:
1703:
2101:
1336:
929:
630:
2425:
2883:
2209:
2135:
1180:
389:
3272:
2759:
1930:
2391:
1621:
2051:
1776:
879:
1851:
3204:
3010:
2849:
1494:
1402:
571:
355:
149:
76:
2415:
1645:
1802:
1575:
1448:
1549:
491:
422:
288:
2222:
2786:
2175:
2155:
2018:
1871:
1723:
1422:
1356:
1087:
1067:
611:
591:
531:
511:
442:
238:
214:
1520:
981:
986:
295:
in real projective space, correspond to two-dimensional subspaces of a four-dimensional vector space). The image of that embedding is the
2851:, the simplest Grassmannian which is not a projective space, and the above reduces to a single equation. Denoting the coordinates of
3425:
3369:
3326:
3121:. The maximal number of algebraically independent relations (on Zariski open sets) is given by the difference of dimension between
3417:
3124:
1246:
1201:
1095:
154:
3020:
1941:
2635:
3457:
2893:
1650:
3413:
2056:
1291:
884:
2157:
just changes the Plücker coordinates by a nonzero scaling factor equal to the determinant of the change of basis matrix
105:
2609:{\displaystyle \sum _{l=1}^{k+1}(-1)^{l}W_{i_{1},\dots ,i_{k-1},j_{l}}W_{j_{1},\dots ,{\hat {j}}_{l},\dots j_{k+1}}=0,}
3452:
1190:
The image under the Plücker embedding satisfies a simple set of homogeneous quadratic relations, usually called the
3353:
3310:
2854:
2180:
2106:
1151:
360:
3209:
2709:
1876:
83:
2344:
1580:
2023:
1728:
3118:
1934:
832:
449:
252:
1807:
1288:
and gives another method of constructing the
Grassmannian. To state the Grassmann–Plücker relations, let
3169:
2975:
2814:
1453:
1361:
536:
320:
114:
41:
3361:
3302:
2396:
1626:
445:
3294:
3111:
In general, many more equations are needed to define the image of the Plücker embedding, as in (
240:. The image is algebraic, consisting of the intersection of a number of quadrics defined by the
2329:{\displaystyle i_{1}<i_{2}<\cdots <i_{k-1},\quad j_{1}<j_{2}<\cdots <j_{k+1}}
1781:
1554:
1427:
3421:
3365:
3349:
3322:
3290:
1525:
306:
248:
463:
394:
255:
3431:
3383:
3332:
3314:
101:
3379:
2764:
3435:
3405:
3387:
3375:
3336:
3298:
292:
86:
2160:
2140:
2003:
1856:
1708:
1407:
1341:
1138:
1072:
1052:
596:
576:
516:
496:
427:
223:
217:
199:
1499:
934:
3446:
296:
1145:
94:
36:
3309:, Encyclopedia of Mathematics and Its Applications, vol. 46 (2nd ed.),
17:
28:
1042:{\displaystyle w_{1}\wedge \cdots \wedge w_{k}\in {\textstyle \bigwedge }^{k}V}
3318:
2393:, the following homogeneous equations are valid, and determine the image of
2103:
under the Plücker map, relative to the standard basis in the exterior space
109:
2177:, and hence just the representative of the projective equivalence class in
1358:-dimensional subspace spanned by the basis represented by column vectors
1243:. This shows that the Grassmannian embeds as an algebraic subvariety of
357:
under the Plücker embedding, relative to the basis in the exterior space
291:
as a way of describing the lines in three-dimensional space (which, as
1092:
This is an embedding of the
Grassmannian into the projectivization
2137:. Changing the basis defining the homogeneous coordinate matrix
1551:
related to each other by right multiplication by an invertible
317:. The homogeneous coordinates of the image of the Grassmannian
2402:
2062:
2035:
1947:
1632:
1297:
890:
714:
3159:{\displaystyle \mathbf {P} ({\textstyle \bigwedge }^{k}V)}
1281:{\displaystyle \mathbf {P} ({\textstyle \bigwedge }^{k}V)}
1236:{\displaystyle \mathbf {P} ({\textstyle \bigwedge }^{k}V)}
1130:{\displaystyle \mathbf {P} ({\textstyle \bigwedge }^{k}V)}
189:{\displaystyle \mathbf {P} ({\textstyle \bigwedge }^{k}V)}
3099:{\displaystyle W_{12}W_{34}-W_{13}W_{24}+W_{14}W_{23}=0.}
1993:{\displaystyle {\mathcal {W}}\sim \in \mathbf {Gr} (k,V)}
3012:
under the Plücker map is defined by the single equation
2699:{\displaystyle j_{1},\dots ,{\hat {j}}_{l}\dots j_{k+1}}
1198:, defining the intersection of a number of quadrics in
3214:
3138:
2860:
2186:
2112:
1450:
matrix of homogeneous coordinates, whose columns are
1260:
1215:
1157:
1109:
1024:
681:
366:
168:
3212:
3172:
3127:
3023:
2978:
2960:{\displaystyle W_{ij}=-W_{ji},\quad 1\leq i,j\leq 4,}
2896:
2857:
2817:
2767:
2712:
2638:
2428:
2399:
2347:
2225:
2183:
2163:
2143:
2109:
2059:
2026:
2006:
1944:
1879:
1859:
1810:
1784:
1731:
1711:
1698:{\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}
1653:
1629:
1583:
1557:
1528:
1502:
1456:
1430:
1410:
1364:
1344:
1294:
1249:
1204:
1154:
1098:
1075:
1055:
989:
937:
887:
835:
628:
599:
579:
539:
519:
499:
466:
430:
397:
363:
323:
258:
226:
202:
157:
117:
44:
2096:{\displaystyle {\mathcal {W}}\in \mathbf {Gr} (k,V)}
1331:{\displaystyle {\mathcal {W}}\in \mathbf {Gr} (k,V)}
924:{\displaystyle {\mathcal {W}}\in \mathbf {Gr} (k,V)}
3360:, Wiley Classics Library (2nd ed.), New York:
983:is the projective equivalence class of the element
3266:
3198:
3158:
3098:
3004:
2959:
2877:
2843:
2780:
2753:
2698:
2608:
2409:
2385:
2328:
2203:
2169:
2149:
2129:
2095:
2045:
2012:
1992:
1924:
1865:
1845:
1796:
1770:
1717:
1697:
1639:
1615:
1569:
1543:
1514:
1488:
1442:
1416:
1396:
1350:
1330:
1280:
1235:
1174:
1129:
1081:
1061:
1041:
975:
923:
873:
816:
605:
585:
565:
525:
505:
485:
436:
416:
383:
349:
282:
232:
208:
188:
143:
70:
2788:omitted. These are generally referred to as the
2020:. They are the linear coordinates of the image
1148:appears as the ring of polynomial functions on
3230:
3217:
1522:of all such homogeneous coordinates matrices
309:generalized Plücker's embedding to arbitrary
8:
2878:{\displaystyle {\textstyle \bigwedge }^{2}V}
2204:{\displaystyle {\textstyle \bigwedge }^{k}V}
2130:{\displaystyle {\textstyle \bigwedge }^{k}V}
1919:
1880:
1175:{\displaystyle {\textstyle \bigwedge }^{k}V}
384:{\displaystyle {\textstyle \bigwedge }^{k}V}
247:The Plücker embedding was first defined by
3267:{\displaystyle {\tbinom {n}{k}}-k(n-k)-1.}
2754:{\displaystyle j_{1},\dots ,\dots j_{k+1}}
1925:{\displaystyle \{W_{i_{1},\dots ,i_{k}}\}}
3229:
3216:
3213:
3211:
3173:
3171:
3144:
3137:
3128:
3126:
3084:
3074:
3061:
3051:
3038:
3028:
3022:
2979:
2977:
2920:
2901:
2895:
2866:
2859:
2856:
2818:
2816:
2772:
2766:
2739:
2717:
2711:
2684:
2671:
2660:
2659:
2643:
2637:
2583:
2567:
2556:
2555:
2539:
2534:
2522:
2503:
2484:
2479:
2469:
2444:
2433:
2427:
2401:
2400:
2398:
2371:
2358:
2346:
2314:
2295:
2282:
2262:
2243:
2230:
2224:
2192:
2185:
2182:
2162:
2142:
2118:
2111:
2108:
2070:
2061:
2060:
2058:
2034:
2033:
2025:
2005:
1967:
1946:
1945:
1943:
1911:
1892:
1887:
1878:
1858:
1834:
1818:
1809:
1783:
1760:
1741:
1736:
1730:
1710:
1683:
1664:
1652:
1631:
1630:
1628:
1590:
1582:
1556:
1527:
1501:
1480:
1461:
1455:
1429:
1409:
1388:
1369:
1363:
1343:
1305:
1296:
1295:
1293:
1266:
1259:
1250:
1248:
1221:
1214:
1205:
1203:
1163:
1156:
1153:
1115:
1108:
1099:
1097:
1074:
1054:
1030:
1023:
1013:
994:
988:
964:
945:
936:
898:
889:
888:
886:
862:
843:
834:
798:
779:
767:
754:
735:
713:
712:
687:
680:
671:
666:
639:
629:
627:
598:
578:
540:
538:
518:
513:-dimensional vector space over the field
498:
477:
465:
429:
408:
396:
372:
365:
362:
324:
322:
257:
225:
201:
174:
167:
158:
156:
118:
116:
45:
43:
3282:
2386:{\displaystyle 1\leq i_{l},j_{m}\leq n}
1616:{\displaystyle g\in \mathbf {GL} (k,K)}
2046:{\displaystyle \iota ({\mathcal {W}})}
1771:{\displaystyle W_{i_{1},\dots ,i_{k}}}
391:corresponding to the natural basis in
241:
7:
2419:
874:{\displaystyle (w_{1},\dots ,w_{k})}
3117:), but these are not, in general,
1846:{\displaystyle (i_{1},\dots i_{k})}
1623:may be identified with the element
613:, the Plücker embedding is the map
3221:
3199:{\displaystyle \mathbf {Gr} (k,V)}
3005:{\displaystyle \mathbf {Gr} (2,V)}
2844:{\displaystyle \mathbf {Gr} (2,V)}
2000:whose homogeneous coordinates are
1489:{\displaystyle W_{1},\dots ,W_{k}}
1397:{\displaystyle W_{1},\dots ,W_{k}}
566:{\displaystyle \mathbf {Gr} (k,V)}
350:{\displaystyle \mathbf {Gr} (k,V)}
144:{\displaystyle \mathbf {Gr} (k,V)}
108:. More precisely, the Plücker map
71:{\displaystyle \mathbf {Gr} (k,V)}
25:
3410:Combinatorial commutative algebra
1873:. Then, up to projectivization,
3358:Principles of algebraic geometry
3177:
3174:
3129:
2983:
2980:
2822:
2819:
2214:For any two ordered sequences:
2074:
2071:
1971:
1968:
1594:
1591:
1309:
1306:
1251:
1206:
1100:
902:
899:
672:
643:
640:
544:
541:
328:
325:
159:
122:
119:
49:
46:
3416:. Vol. 227. New York, NY:
2932:
2277:
1804:matrix whose rows are the rows
100:, either real or complex, in a
3255:
3243:
3193:
3181:
3153:
3133:
2999:
2987:
2838:
2826:
2665:
2561:
2466:
2456:
2410:{\displaystyle {\mathcal {W}}}
2090:
2078:
2040:
2030:
1987:
1975:
1961:
1955:
1840:
1811:
1640:{\displaystyle {\mathcal {W}}}
1610:
1598:
1509:
1503:
1325:
1313:
1275:
1255:
1230:
1210:
1124:
1104:
970:
938:
918:
906:
868:
836:
804:
772:
769:
760:
728:
696:
676:
668:
659:
647:
560:
548:
344:
332:
183:
163:
138:
126:
65:
53:
1:
3414:Graduate Texts in Mathematics
1496:. Then the equivalence class
881:is a basis for the element
106:projective algebraic variety
104:, thereby realizing it as a
3113:
1647:. For any ordered sequence
1196:Grassmann–Plücker relations
3474:
3311:Cambridge University Press
1778:be the determinant of the
593:-dimensional subspaces of
151:into the projectivization
3119:algebraically independent
1797:{\displaystyle k\times k}
1570:{\displaystyle k\times k}
1443:{\displaystyle n\times k}
3319:10.1017/CBO9780511586507
1544:{\displaystyle Wg\sim W}
242:§ Plücker relations
2417:under the Plücker map:
486:{\displaystyle V=K^{n}}
417:{\displaystyle V=K^{n}}
283:{\displaystyle k=2,n=4}
3268:
3200:
3160:
3100:
3006:
2961:
2879:
2845:
2782:
2755:
2700:
2610:
2455:
2411:
2387:
2330:
2205:
2171:
2151:
2131:
2097:
2047:
2014:
1994:
1926:
1867:
1847:
1798:
1772:
1719:
1699:
1641:
1617:
1571:
1545:
1516:
1490:
1444:
1418:
1398:
1352:
1332:
1282:
1237:
1176:
1131:
1083:
1063:
1043:
977:
925:
875:
818:
607:
587:
567:
527:
507:
487:
438:
418:
385:
351:
284:
234:
210:
190:
145:
72:
3458:Differential geometry
3362:John Wiley & Sons
3269:
3201:
3161:
3101:
3007:
2962:
2880:
2846:
2783:
2781:{\displaystyle j_{l}}
2756:
2706:denotes the sequence
2701:
2611:
2429:
2412:
2388:
2341:of positive integers
2331:
2206:
2172:
2152:
2132:
2098:
2048:
2015:
1995:
1927:
1868:
1848:
1799:
1773:
1720:
1700:
1642:
1618:
1572:
1546:
1517:
1491:
1445:
1419:
1399:
1353:
1333:
1283:
1238:
1177:
1132:
1084:
1069:th exterior power of
1064:
1044:
978:
926:
876:
819:
608:
588:
568:
528:
508:
488:
439:
419:
386:
352:
285:
235:
211:
191:
146:
78:, whose elements are
73:
3210:
3170:
3125:
3021:
2976:
2894:
2855:
2815:
2765:
2710:
2636:
2426:
2397:
2345:
2223:
2181:
2161:
2141:
2107:
2057:
2024:
2004:
1942:
1877:
1857:
1808:
1782:
1729:
1709:
1651:
1627:
1581:
1555:
1526:
1500:
1454:
1428:
1408:
1362:
1342:
1292:
1247:
1202:
1152:
1096:
1073:
1053:
987:
935:
885:
833:
626:
597:
577:
573:the Grassmannian of
537:
517:
497:
464:
428:
395:
361:
321:
256:
224:
200:
155:
115:
42:
3295:Las Vergnas, Michel
1935:Plücker coordinates
450:Plücker coordinates
3453:Algebraic geometry
3350:Griffiths, Phillip
3264:
3235:
3196:
3156:
3142:
3096:
3002:
2957:
2875:
2864:
2841:
2778:
2751:
2696:
2606:
2407:
2383:
2326:
2201:
2190:
2167:
2147:
2127:
2116:
2093:
2043:
2010:
1990:
1922:
1863:
1843:
1794:
1768:
1715:
1695:
1637:
1613:
1567:
1541:
1512:
1486:
1440:
1414:
1394:
1348:
1328:
1278:
1264:
1233:
1219:
1172:
1161:
1127:
1113:
1079:
1059:
1039:
1028:
973:
921:
871:
814:
812:
685:
603:
583:
563:
523:
503:
483:
434:
414:
381:
370:
347:
280:
230:
206:
186:
172:
141:
68:
3307:Oriented matroids
3228:
2790:Plücker relations
2668:
2630:
2629:
2564:
2170:{\displaystyle g}
2150:{\displaystyle M}
2013:{\displaystyle W}
1866:{\displaystyle W}
1718:{\displaystyle k}
1417:{\displaystyle W}
1351:{\displaystyle k}
1192:Plücker relations
1186:Plücker relations
1082:{\displaystyle V}
1062:{\displaystyle k}
606:{\displaystyle V}
586:{\displaystyle k}
526:{\displaystyle K}
506:{\displaystyle n}
437:{\displaystyle K}
307:Hermann Grassmann
233:{\displaystyle V}
209:{\displaystyle k}
18:Plücker relations
16:(Redirected from
3465:
3439:
3406:Sturmfels, Bernd
3391:
3390:
3346:
3340:
3339:
3299:Sturmfels, Bernd
3287:
3273:
3271:
3270:
3265:
3236:
3234:
3233:
3220:
3205:
3203:
3202:
3197:
3180:
3165:
3163:
3162:
3157:
3149:
3148:
3143:
3132:
3105:
3103:
3102:
3097:
3089:
3088:
3079:
3078:
3066:
3065:
3056:
3055:
3043:
3042:
3033:
3032:
3011:
3009:
3008:
3003:
2986:
2966:
2964:
2963:
2958:
2928:
2927:
2909:
2908:
2884:
2882:
2881:
2876:
2871:
2870:
2865:
2850:
2848:
2847:
2842:
2825:
2810:
2803:
2787:
2785:
2784:
2779:
2777:
2776:
2760:
2758:
2757:
2752:
2750:
2749:
2722:
2721:
2705:
2703:
2702:
2697:
2695:
2694:
2676:
2675:
2670:
2669:
2661:
2648:
2647:
2624:
2615:
2613:
2612:
2607:
2596:
2595:
2594:
2593:
2572:
2571:
2566:
2565:
2557:
2544:
2543:
2529:
2528:
2527:
2526:
2514:
2513:
2489:
2488:
2474:
2473:
2454:
2443:
2420:
2416:
2414:
2413:
2408:
2406:
2405:
2392:
2390:
2389:
2384:
2376:
2375:
2363:
2362:
2335:
2333:
2332:
2327:
2325:
2324:
2300:
2299:
2287:
2286:
2273:
2272:
2248:
2247:
2235:
2234:
2210:
2208:
2207:
2202:
2197:
2196:
2191:
2176:
2174:
2173:
2168:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2123:
2122:
2117:
2102:
2100:
2099:
2094:
2077:
2066:
2065:
2052:
2050:
2049:
2044:
2039:
2038:
2019:
2017:
2016:
2011:
1999:
1997:
1996:
1991:
1974:
1951:
1950:
1931:
1929:
1928:
1923:
1918:
1917:
1916:
1915:
1897:
1896:
1872:
1870:
1869:
1864:
1852:
1850:
1849:
1844:
1839:
1838:
1823:
1822:
1803:
1801:
1800:
1795:
1777:
1775:
1774:
1769:
1767:
1766:
1765:
1764:
1746:
1745:
1724:
1722:
1721:
1716:
1704:
1702:
1701:
1696:
1688:
1687:
1669:
1668:
1646:
1644:
1643:
1638:
1636:
1635:
1622:
1620:
1619:
1614:
1597:
1576:
1574:
1573:
1568:
1550:
1548:
1547:
1542:
1521:
1519:
1518:
1515:{\displaystyle }
1513:
1495:
1493:
1492:
1487:
1485:
1484:
1466:
1465:
1449:
1447:
1446:
1441:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1393:
1392:
1374:
1373:
1357:
1355:
1354:
1349:
1337:
1335:
1334:
1329:
1312:
1301:
1300:
1287:
1285:
1284:
1279:
1271:
1270:
1265:
1254:
1242:
1240:
1239:
1234:
1226:
1225:
1220:
1209:
1181:
1179:
1178:
1173:
1168:
1167:
1162:
1136:
1134:
1133:
1128:
1120:
1119:
1114:
1103:
1088:
1086:
1085:
1080:
1068:
1066:
1065:
1060:
1048:
1046:
1045:
1040:
1035:
1034:
1029:
1018:
1017:
999:
998:
982:
980:
979:
976:{\displaystyle }
974:
969:
968:
950:
949:
930:
928:
927:
922:
905:
894:
893:
880:
878:
877:
872:
867:
866:
848:
847:
823:
821:
820:
815:
813:
803:
802:
784:
783:
768:
759:
758:
740:
739:
718:
717:
692:
691:
686:
675:
667:
646:
612:
610:
609:
604:
592:
590:
589:
584:
572:
570:
569:
564:
547:
532:
530:
529:
524:
512:
510:
509:
504:
492:
490:
489:
484:
482:
481:
443:
441:
440:
435:
423:
421:
420:
415:
413:
412:
390:
388:
387:
382:
377:
376:
371:
356:
354:
353:
348:
331:
293:projective lines
289:
287:
286:
281:
239:
237:
236:
231:
215:
213:
212:
207:
195:
193:
192:
187:
179:
178:
173:
162:
150:
148:
147:
142:
125:
102:projective space
77:
75:
74:
69:
52:
21:
3473:
3472:
3468:
3467:
3466:
3464:
3463:
3462:
3443:
3442:
3428:
3418:Springer-Verlag
3403:
3400:
3398:Further reading
3395:
3394:
3372:
3364:, p. 211,
3348:
3347:
3343:
3329:
3303:Ziegler, Günter
3301:; White, Neil;
3291:Björner, Anders
3289:
3288:
3284:
3279:
3215:
3208:
3207:
3168:
3167:
3136:
3123:
3122:
3080:
3070:
3057:
3047:
3034:
3024:
3019:
3018:
2974:
2973:
2916:
2897:
2892:
2891:
2858:
2853:
2852:
2813:
2812:
2805:
2797:
2795:
2768:
2763:
2762:
2735:
2713:
2708:
2707:
2680:
2658:
2639:
2634:
2633:
2622:
2579:
2554:
2535:
2530:
2518:
2499:
2480:
2475:
2465:
2424:
2423:
2395:
2394:
2367:
2354:
2343:
2342:
2310:
2291:
2278:
2258:
2239:
2226:
2221:
2220:
2184:
2179:
2178:
2159:
2158:
2139:
2138:
2110:
2105:
2104:
2055:
2054:
2022:
2021:
2002:
2001:
1940:
1939:
1938:of the element
1907:
1888:
1883:
1875:
1874:
1855:
1854:
1830:
1814:
1806:
1805:
1780:
1779:
1756:
1737:
1732:
1727:
1726:
1707:
1706:
1679:
1660:
1649:
1648:
1625:
1624:
1579:
1578:
1553:
1552:
1524:
1523:
1498:
1497:
1476:
1457:
1452:
1451:
1426:
1425:
1406:
1405:
1384:
1365:
1360:
1359:
1340:
1339:
1290:
1289:
1258:
1245:
1244:
1213:
1200:
1199:
1188:
1155:
1150:
1149:
1107:
1094:
1093:
1071:
1070:
1051:
1050:
1022:
1009:
990:
985:
984:
960:
941:
933:
932:
883:
882:
858:
839:
831:
830:
811:
810:
794:
775:
763:
750:
731:
703:
702:
679:
662:
624:
623:
595:
594:
575:
574:
535:
534:
515:
514:
495:
494:
473:
462:
461:
458:
426:
425:
404:
393:
392:
364:
359:
358:
319:
318:
254:
253:
222:
221:
198:
197:
166:
153:
152:
113:
112:
40:
39:
23:
22:
15:
12:
11:
5:
3471:
3469:
3461:
3460:
3455:
3445:
3444:
3441:
3440:
3426:
3404:Miller, Ezra;
3399:
3396:
3393:
3392:
3370:
3354:Harris, Joseph
3341:
3327:
3313:, p. 79,
3281:
3280:
3278:
3275:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3232:
3227:
3224:
3219:
3195:
3192:
3189:
3186:
3183:
3179:
3176:
3155:
3152:
3147:
3141:
3135:
3131:
3109:
3108:
3107:
3106:
3095:
3092:
3087:
3083:
3077:
3073:
3069:
3064:
3060:
3054:
3050:
3046:
3041:
3037:
3031:
3027:
3001:
2998:
2995:
2992:
2989:
2985:
2982:
2970:
2969:
2968:
2967:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2931:
2926:
2923:
2919:
2915:
2912:
2907:
2904:
2900:
2874:
2869:
2863:
2840:
2837:
2834:
2831:
2828:
2824:
2821:
2775:
2771:
2761:with the term
2748:
2745:
2742:
2738:
2734:
2731:
2728:
2725:
2720:
2716:
2693:
2690:
2687:
2683:
2679:
2674:
2667:
2664:
2657:
2654:
2651:
2646:
2642:
2628:
2627:
2618:
2616:
2605:
2602:
2599:
2592:
2589:
2586:
2582:
2578:
2575:
2570:
2563:
2560:
2553:
2550:
2547:
2542:
2538:
2533:
2525:
2521:
2517:
2512:
2509:
2506:
2502:
2498:
2495:
2492:
2487:
2483:
2478:
2472:
2468:
2464:
2461:
2458:
2453:
2450:
2447:
2442:
2439:
2436:
2432:
2404:
2382:
2379:
2374:
2370:
2366:
2361:
2357:
2353:
2350:
2339:
2338:
2337:
2336:
2323:
2320:
2317:
2313:
2309:
2306:
2303:
2298:
2294:
2290:
2285:
2281:
2276:
2271:
2268:
2265:
2261:
2257:
2254:
2251:
2246:
2242:
2238:
2233:
2229:
2200:
2195:
2189:
2166:
2146:
2126:
2121:
2115:
2092:
2089:
2086:
2083:
2080:
2076:
2073:
2069:
2064:
2042:
2037:
2032:
2029:
2009:
1989:
1986:
1983:
1980:
1977:
1973:
1970:
1966:
1963:
1960:
1957:
1954:
1949:
1921:
1914:
1910:
1906:
1903:
1900:
1895:
1891:
1886:
1882:
1862:
1842:
1837:
1833:
1829:
1826:
1821:
1817:
1813:
1793:
1790:
1787:
1763:
1759:
1755:
1752:
1749:
1744:
1740:
1735:
1725:integers, let
1714:
1694:
1691:
1686:
1682:
1678:
1675:
1672:
1667:
1663:
1659:
1656:
1634:
1612:
1609:
1606:
1603:
1600:
1596:
1593:
1589:
1586:
1566:
1563:
1560:
1540:
1537:
1534:
1531:
1511:
1508:
1505:
1483:
1479:
1475:
1472:
1469:
1464:
1460:
1439:
1436:
1433:
1413:
1391:
1387:
1383:
1380:
1377:
1372:
1368:
1347:
1327:
1324:
1321:
1318:
1315:
1311:
1308:
1304:
1299:
1277:
1274:
1269:
1263:
1257:
1253:
1232:
1229:
1224:
1218:
1212:
1208:
1187:
1184:
1171:
1166:
1160:
1139:linear algebra
1126:
1123:
1118:
1112:
1106:
1102:
1078:
1058:
1038:
1033:
1027:
1021:
1016:
1012:
1008:
1005:
1002:
997:
993:
972:
967:
963:
959:
956:
953:
948:
944:
940:
920:
917:
914:
911:
908:
904:
901:
897:
892:
870:
865:
861:
857:
854:
851:
846:
842:
838:
827:
826:
825:
824:
809:
806:
801:
797:
793:
790:
787:
782:
778:
774:
771:
766:
764:
762:
757:
753:
749:
746:
743:
738:
734:
730:
727:
724:
721:
716:
711:
708:
705:
704:
701:
698:
695:
690:
684:
678:
674:
670:
665:
663:
661:
658:
655:
652:
649:
645:
642:
638:
635:
632:
631:
602:
582:
562:
559:
556:
553:
550:
546:
543:
522:
502:
480:
476:
472:
469:
457:
454:
433:
411:
407:
403:
400:
380:
375:
369:
346:
343:
340:
337:
334:
330:
327:
279:
276:
273:
270:
267:
264:
261:
249:Julius Plücker
229:
218:exterior power
205:
185:
182:
177:
171:
165:
161:
140:
137:
134:
131:
128:
124:
121:
67:
64:
61:
58:
55:
51:
48:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3470:
3459:
3456:
3454:
3451:
3450:
3448:
3437:
3433:
3429:
3427:0-387-23707-0
3423:
3419:
3415:
3411:
3407:
3402:
3401:
3397:
3389:
3385:
3381:
3377:
3373:
3371:0-471-05059-8
3367:
3363:
3359:
3355:
3351:
3345:
3342:
3338:
3334:
3330:
3328:0-521-77750-X
3324:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3286:
3283:
3276:
3274:
3261:
3258:
3252:
3249:
3246:
3240:
3237:
3225:
3222:
3190:
3187:
3184:
3150:
3145:
3139:
3120:
3116:
3115:
3093:
3090:
3085:
3081:
3075:
3071:
3067:
3062:
3058:
3052:
3048:
3044:
3039:
3035:
3029:
3025:
3017:
3016:
3015:
3014:
3013:
2996:
2993:
2990:
2972:the image of
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2929:
2924:
2921:
2917:
2913:
2910:
2905:
2902:
2898:
2890:
2889:
2888:
2887:
2886:
2872:
2867:
2861:
2835:
2832:
2829:
2808:
2801:
2793:
2791:
2773:
2769:
2746:
2743:
2740:
2736:
2732:
2729:
2726:
2723:
2718:
2714:
2691:
2688:
2685:
2681:
2677:
2672:
2662:
2655:
2652:
2649:
2644:
2640:
2626:
2619:
2617:
2603:
2600:
2597:
2590:
2587:
2584:
2580:
2576:
2573:
2568:
2558:
2551:
2548:
2545:
2540:
2536:
2531:
2523:
2519:
2515:
2510:
2507:
2504:
2500:
2496:
2493:
2490:
2485:
2481:
2476:
2470:
2462:
2459:
2451:
2448:
2445:
2440:
2437:
2434:
2430:
2422:
2421:
2418:
2380:
2377:
2372:
2368:
2364:
2359:
2355:
2351:
2348:
2321:
2318:
2315:
2311:
2307:
2304:
2301:
2296:
2292:
2288:
2283:
2279:
2274:
2269:
2266:
2263:
2259:
2255:
2252:
2249:
2244:
2240:
2236:
2231:
2227:
2219:
2218:
2217:
2216:
2215:
2212:
2198:
2193:
2187:
2164:
2144:
2124:
2119:
2113:
2087:
2084:
2081:
2067:
2027:
2007:
1984:
1981:
1978:
1964:
1958:
1952:
1937:
1936:
1912:
1908:
1904:
1901:
1898:
1893:
1889:
1884:
1860:
1835:
1831:
1827:
1824:
1819:
1815:
1791:
1788:
1785:
1761:
1757:
1753:
1750:
1747:
1742:
1738:
1733:
1712:
1692:
1689:
1684:
1680:
1676:
1673:
1670:
1665:
1661:
1657:
1654:
1607:
1604:
1601:
1587:
1584:
1564:
1561:
1558:
1538:
1535:
1532:
1529:
1506:
1481:
1477:
1473:
1470:
1467:
1462:
1458:
1437:
1434:
1431:
1411:
1389:
1385:
1381:
1378:
1375:
1370:
1366:
1345:
1322:
1319:
1316:
1302:
1272:
1267:
1261:
1227:
1222:
1216:
1197:
1193:
1185:
1183:
1169:
1164:
1158:
1147:
1142:
1140:
1121:
1116:
1110:
1090:
1076:
1056:
1036:
1031:
1025:
1019:
1014:
1010:
1006:
1003:
1000:
995:
991:
965:
961:
957:
954:
951:
946:
942:
915:
912:
909:
895:
863:
859:
855:
852:
849:
844:
840:
807:
799:
795:
791:
788:
785:
780:
776:
765:
755:
751:
747:
744:
741:
736:
732:
725:
722:
719:
709:
706:
699:
693:
688:
682:
664:
656:
653:
650:
636:
633:
622:
621:
620:
619:
618:
616:
600:
580:
557:
554:
551:
520:
500:
478:
474:
470:
467:
455:
453:
451:
448:) are called
447:
431:
409:
405:
401:
398:
378:
373:
367:
341:
338:
335:
316:
312:
308:
304:
302:
298:
297:Klein quadric
294:
290:
277:
274:
271:
268:
265:
262:
259:
251:in the case
250:
245:
244:(see below).
243:
227:
219:
203:
180:
175:
169:
135:
132:
129:
111:
107:
103:
99:
96:
93:-dimensional
92:
88:
85:
81:
62:
59:
56:
38:
34:
30:
19:
3409:
3357:
3344:
3306:
3285:
3112:
3110:
2971:
2806:
2799:
2794:
2789:
2631:
2620:
2340:
2213:
1933:
1195:
1191:
1189:
1146:bracket ring
1143:
1091:
828:
614:
460:Denoting by
459:
444:is the base
314:
310:
305:
300:
246:
97:
95:vector space
90:
79:
37:Grassmannian
32:
26:
3206:, which is
617:defined by
84:dimensional
35:embeds the
33:Plücker map
29:mathematics
3447:Categories
3436:1090.13001
3388:0836.14001
3337:0944.52006
3277:References
456:Definition
3259:−
3250:−
3238:−
3140:⋀
3045:−
2949:≤
2937:≤
2914:−
2862:⋀
2811:, we get
2733:…
2727:…
2678:…
2666:^
2653:…
2577:…
2562:^
2549:…
2508:−
2494:…
2460:−
2431:∑
2378:≤
2352:≤
2305:⋯
2267:−
2253:⋯
2188:⋀
2114:⋀
2068:∈
2028:ι
1965:∈
1953:∼
1902:…
1828:…
1789:×
1751:…
1690:≤
1674:⋯
1658:≤
1588:∈
1562:×
1536:∼
1471:…
1435:×
1379:…
1303:∈
1262:⋀
1217:⋀
1159:⋀
1111:⋀
1026:⋀
1020:∈
1007:∧
1004:⋯
1001:∧
958:∧
955:⋯
952:∧
896:∈
853:…
792:∧
789:⋯
786:∧
770:↦
745:…
726:
710::
707:ι
683:⋀
669:→
637::
634:ι
533:, and by
368:⋀
170:⋀
87:subspaces
3408:(2005).
3356:(1994),
3305:(1999),
1932:are the
1424:be the
3380:1288523
1577:matrix
1404:. Let
1338:be the
1049:of the
424:(where
196:of the
31:, the
3434:
3424:
3386:
3378:
3368:
3335:
3325:
2632:where
829:where
110:embeds
89:of an
2802:) = 4
2796:When
1194:, or
446:field
3422:ISBN
3366:ISBN
3323:ISBN
3166:and
2885:by
2804:and
2798:dim(
2308:<
2302:<
2289:<
2256:<
2250:<
2237:<
2053:of
1677:<
1671:<
1144:The
931:and
723:span
493:the
313:and
216:-th
3432:Zbl
3384:Zbl
3333:Zbl
3315:doi
2809:= 2
2211:.
1853:of
1705:of
299:in
220:of
27:In
3449::
3430:.
3420:.
3412:.
3382:,
3376:MR
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