400:
1590:
constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick
1449:. Every analytically integral element is integral in the Lie algebra sense, but there may be integral elements in the Lie algebra sense that are not analytically integral. This distinction reflects the fact that if
1246:
1107:
1598:
approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representations. This approach is essentially due to H. Weyl and works quite well for classical
529:
599:
487:
335:
1320:
1359:
1292:. The theory then proceeds in much the same way as in the Lie algebra case, with one crucial difference: the notion of integrality is different. Specifically, we say that an element
673:
562:
122:
1491:
1270:
1199:
1131:
1060:
928:
813:
274:
247:
149:
53:
743:
1404:
994:
974:
837:
769:
693:
622:
203:
423:
450:
857:
717:
642:
347:
1554:
1531:
1511:
1467:
1447:
1427:
1290:
1175:
1151:
1032:
948:
904:
877:
789:
294:
189:
169:
80:
1954:
1683:"Reviewed work: Matrix Groups: An Introduction to Lie Group Theory, Andrew Baker; Lie Groups: An Introduction through Linear Groups, Wulf Rossmann"
1000:
The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.
1920:
1860:
1815:
1204:
1065:
1895:
171:. The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If
1587:
1949:
1583:
contains the highest weight theorem. This is the approach taken in many standard textbooks (e.g., Humphreys and Part II of Hall).
1803:
1939:
1832:
1619:
749:
220:
496:
1844:
1807:
567:
455:
303:
1836:
1609:
25:
1295:
127:
from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of
1332:
1556:
is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."
1573:
1569:
29:
647:
1944:
953:
If two finite-dimensional irreducible representations have the same highest weight, they are isomorphic.
17:
534:
88:
1472:
1251:
1180:
1112:
1041:
909:
794:
255:
228:
221:
Weight (representation theory) § Weights in the representation theory of semisimple Lie algebras
130:
34:
722:
1370:
1904:
1702:
1916:
1891:
1874:
1856:
1811:
979:
959:
822:
754:
745:
is expressible as a linear combination of positive roots with non-negative real coefficients.
678:
607:
250:
408:
1848:
1694:
1595:
395:{\displaystyle 2{\frac {\langle \lambda ,\alpha \rangle }{\langle \alpha ,\alpha \rangle }}}
1870:
1825:
428:
1866:
1821:
842:
702:
627:
1539:
1516:
1496:
1452:
1432:
1412:
1275:
1160:
1136:
1017:
933:
889:
862:
774:
279:
195:
174:
154:
65:
1933:
1909:
1614:
1533:
is simply connected, the notions of "integral" and "analytically integral" coincide.
1154:
1035:
1009:
56:
1580:
976:, there exists a finite-dimensional irreducible representation with highest weight
199:
1797:
1568:
Hermann Weyl's original proof from the compact group point of view, based on the
297:
198:
in his 1913 paper. The version of the theorem for a compact Lie group is due to
1852:
1878:
60:
1010:
Compact group § Representation theory of a connected compact Lie group
950:
has a unique highest weight, and this highest weight is dominant integral.
1890:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
1888:
Lie groups, Lie algebras, and representations: An elementary introduction
1706:
1682:
1847:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
1698:
1241:{\displaystyle {\mathfrak {h}}:={\mathfrak {t}}+i{\mathfrak {t}}}
1102:{\displaystyle {\mathfrak {g}}:={\mathfrak {k}}+i{\mathfrak {k}}}
1620:
Weights in the representation theory of semisimple Lie algebras
1610:
Classifying finite-dimensional representations of Lie algebras
249:
be a finite-dimensional semisimple complex Lie algebra with
1536:
The theorem of the highest weight for representations of
1469:
is not simply connected, there may be representations of
1911:
Introduction to Lie
Algebras and Representation Theory
1615:
Representation theory of a connected compact Lie group
906:
is a finite-dimensional irreducible representation of
524:{\displaystyle \langle \lambda ,\alpha \rangle \geq 0}
1542:
1519:
1499:
1475:
1455:
1435:
1415:
1373:
1335:
1298:
1278:
1254:
1207:
1183:
1163:
1139:
1115:
1068:
1044:
1020:
982:
962:
936:
912:
892:
865:
845:
825:
797:
777:
757:
725:
705:
681:
650:
630:
610:
570:
537:
499:
458:
431:
411:
350:
306:
282:
258:
231:
177:
157:
133:
91:
68:
55:. There is a closely related theorem classifying the
37:
1908:
1548:
1525:
1505:
1485:
1461:
1441:
1421:
1398:
1353:
1314:
1284:
1264:
1240:
1193:
1169:
1145:
1125:
1101:
1054:
1026:
988:
968:
942:
922:
898:
871:
851:
831:
807:
783:
763:
737:
711:
687:
667:
636:
616:
593:
556:
523:
481:
444:
417:
394:
329:
288:
268:
241:
191:is simply connected, this distinction disappears.
183:
163:
143:
116:
74:
47:
604:if it is both dominant and integral. Finally, if
204:representation theory of semisimple Lie algebras
882:The theorem of the highest weight then states:
594:{\displaystyle \lambda \in {\mathfrak {h}}^{*}}
482:{\displaystyle \lambda \in {\mathfrak {h}}^{*}}
330:{\displaystyle \lambda \in {\mathfrak {h}}^{*}}
82:. The theorem states that there is a bijection
1649:
1647:
1272:, and we may form the associated root system
452:of positive roots and we say that an element
8:
1348:
1336:
512:
500:
386:
374:
369:
357:
1315:{\displaystyle \lambda \in {\mathfrak {h}}}
1354:{\displaystyle \langle \lambda ,H\rangle }
202:. The theorem is one of the key pieces of
1541:
1518:
1498:
1493:that do not come from representations of
1477:
1476:
1474:
1454:
1434:
1414:
1378:
1372:
1334:
1306:
1305:
1297:
1277:
1256:
1255:
1253:
1232:
1231:
1219:
1218:
1209:
1208:
1206:
1185:
1184:
1182:
1162:
1138:
1117:
1116:
1114:
1093:
1092:
1080:
1079:
1070:
1069:
1067:
1046:
1045:
1043:
1019:
981:
961:
935:
914:
913:
911:
891:
864:
844:
824:
799:
798:
796:
776:
756:
724:
704:
680:
659:
653:
652:
649:
629:
609:
585:
579:
578:
569:
548:
536:
498:
473:
467:
466:
457:
436:
430:
410:
354:
349:
321:
315:
314:
305:
281:
260:
259:
257:
233:
232:
230:
176:
156:
135:
134:
132:
105:
90:
67:
39:
38:
36:
1638:
1631:
1664:
1662:
1841:Representation theory. A first course
194:The theorem was originally proved by
7:
1779:
1767:
1755:
1743:
1731:
1719:
1668:
1653:
1478:
1307:
1257:
1233:
1220:
1210:
1186:
1118:
1094:
1081:
1071:
1047:
956:For each dominant integral element
915:
800:
668:{\displaystyle {\mathfrak {h}}^{*}}
654:
580:
468:
316:
261:
234:
136:
40:
1806:, vol. 11, Providence, R.I.:
839:is higher than every other weight
14:
1955:Theorems in representation theory
1687:The American Mathematical Monthly
1564:There are at least four proofs:
557:{\displaystyle \alpha \in R^{+}}
117:{\displaystyle \lambda \mapsto }
1804:Graduate Studies in Mathematics
1486:{\displaystyle {\mathfrak {g}}}
1265:{\displaystyle {\mathfrak {g}}}
1194:{\displaystyle {\mathfrak {t}}}
1126:{\displaystyle {\mathfrak {g}}}
1055:{\displaystyle {\mathfrak {k}}}
923:{\displaystyle {\mathfrak {g}}}
808:{\displaystyle {\mathfrak {g}}}
269:{\displaystyle {\mathfrak {h}}}
242:{\displaystyle {\mathfrak {g}}}
144:{\displaystyle {\mathfrak {g}}}
48:{\displaystyle {\mathfrak {g}}}
20:, a branch of mathematics, the
300:. We then say that an element
111:
98:
95:
1:
1845:Graduate Texts in Mathematics
1808:American Mathematical Society
738:{\displaystyle \lambda -\mu }
22:theorem of the highest weight
1399:{\displaystyle e^{2\pi H}=I}
405:is an integer for each root
1429:is the identity element of
1109:be the complexification of
57:irreducible representations
26:irreducible representations
1971:
1796:Dixmier, Jacques (1996) ,
1248:is a Cartan subalgebra of
1007:
218:
1853:10.1007/978-1-4612-0979-9
1513:. On the other hand, if
989:{\displaystyle \lambda }
969:{\displaystyle \lambda }
832:{\displaystyle \lambda }
764:{\displaystyle \lambda }
688:{\displaystyle \lambda }
617:{\displaystyle \lambda }
425:. Next, we choose a set
1950:Theorems about algebras
1886:Hall, Brian C. (2015),
1588:Borel–Weil–Bott theorem
1364:is an integer whenever
418:{\displaystyle \alpha }
59:of a connected compact
1570:Weyl character formula
1550:
1527:
1507:
1487:
1463:
1443:
1423:
1400:
1355:
1316:
1286:
1266:
1242:
1195:
1171:
1147:
1127:
1103:
1056:
1028:
1004:The compact group case
990:
970:
944:
924:
900:
873:
853:
833:
809:
785:
765:
739:
713:
689:
669:
638:
618:
595:
558:
525:
483:
446:
419:
396:
331:
290:
270:
243:
185:
165:
145:
118:
76:
49:
30:semisimple Lie algebra
1940:Representation theory
1681:Knapp, A. W. (2003).
1551:
1528:
1508:
1488:
1464:
1444:
1424:
1401:
1356:
1324:analytically integral
1317:
1287:
1267:
1243:
1196:
1172:
1148:
1128:
1104:
1057:
1029:
991:
971:
945:
925:
901:
874:
854:
834:
810:
786:
766:
740:
714:
690:
670:
639:
619:
596:
559:
526:
484:
447:
445:{\displaystyle R^{+}}
420:
397:
332:
291:
271:
244:
186:
166:
146:
119:
77:
50:
18:representation theory
1656:Theorems 9.4 and 9.5
1540:
1517:
1497:
1473:
1453:
1433:
1413:
1371:
1333:
1296:
1276:
1252:
1205:
1181:
1161:
1137:
1113:
1066:
1042:
1018:
980:
960:
934:
910:
890:
863:
852:{\displaystyle \mu }
843:
823:
795:
775:
771:of a representation
755:
723:
712:{\displaystyle \mu }
703:
679:
648:
637:{\displaystyle \mu }
628:
608:
568:
535:
497:
456:
429:
409:
348:
304:
280:
256:
229:
175:
155:
131:
89:
66:
35:
1905:Humphreys, James E.
1799:Enveloping algebras
1596:invariant theoretic
1574:Peter–Weyl theorem
1546:
1523:
1503:
1483:
1459:
1439:
1419:
1396:
1351:
1312:
1282:
1262:
1238:
1191:
1167:
1143:
1123:
1099:
1052:
1024:
986:
966:
940:
920:
896:
869:
849:
829:
805:
781:
761:
735:
709:
685:
665:
634:
614:
591:
554:
521:
479:
442:
415:
392:
327:
296:be the associated
286:
266:
239:
181:
161:
141:
114:
72:
45:
1922:978-0-387-90053-7
1862:978-0-387-97495-8
1817:978-0-8218-0560-2
1549:{\displaystyle K}
1526:{\displaystyle K}
1506:{\displaystyle K}
1462:{\displaystyle K}
1442:{\displaystyle K}
1422:{\displaystyle I}
1285:{\displaystyle R}
1177:with Lie algebra
1170:{\displaystyle K}
1146:{\displaystyle T}
1038:with Lie algebra
1036:compact Lie group
1027:{\displaystyle K}
943:{\displaystyle V}
899:{\displaystyle V}
872:{\displaystyle V}
815:is then called a
784:{\displaystyle V}
602:dominant integral
390:
289:{\displaystyle R}
251:Cartan subalgebra
184:{\displaystyle K}
164:{\displaystyle K}
75:{\displaystyle K}
1962:
1925:
1914:
1900:
1882:
1828:
1783:
1777:
1771:
1765:
1759:
1758:Proposition 12.7
1753:
1747:
1741:
1735:
1729:
1723:
1717:
1711:
1710:
1678:
1672:
1666:
1657:
1651:
1642:
1641:, Theorem 7.2.6.
1636:
1555:
1553:
1552:
1547:
1532:
1530:
1529:
1524:
1512:
1510:
1509:
1504:
1492:
1490:
1489:
1484:
1482:
1481:
1468:
1466:
1465:
1460:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1405:
1403:
1402:
1397:
1389:
1388:
1360:
1358:
1357:
1352:
1321:
1319:
1318:
1313:
1311:
1310:
1291:
1289:
1288:
1283:
1271:
1269:
1268:
1263:
1261:
1260:
1247:
1245:
1244:
1239:
1237:
1236:
1224:
1223:
1214:
1213:
1200:
1198:
1197:
1192:
1190:
1189:
1176:
1174:
1173:
1168:
1152:
1150:
1149:
1144:
1132:
1130:
1129:
1124:
1122:
1121:
1108:
1106:
1105:
1100:
1098:
1097:
1085:
1084:
1075:
1074:
1061:
1059:
1058:
1053:
1051:
1050:
1033:
1031:
1030:
1025:
995:
993:
992:
987:
975:
973:
972:
967:
949:
947:
946:
941:
929:
927:
926:
921:
919:
918:
905:
903:
902:
897:
878:
876:
875:
870:
858:
856:
855:
850:
838:
836:
835:
830:
814:
812:
811:
806:
804:
803:
790:
788:
787:
782:
770:
768:
767:
762:
744:
742:
741:
736:
718:
716:
715:
710:
694:
692:
691:
686:
674:
672:
671:
666:
664:
663:
658:
657:
643:
641:
640:
635:
623:
621:
620:
615:
600:
598:
597:
592:
590:
589:
584:
583:
563:
561:
560:
555:
553:
552:
530:
528:
527:
522:
488:
486:
485:
480:
478:
477:
472:
471:
451:
449:
448:
443:
441:
440:
424:
422:
421:
416:
401:
399:
398:
393:
391:
389:
372:
355:
336:
334:
333:
328:
326:
325:
320:
319:
295:
293:
292:
287:
275:
273:
272:
267:
265:
264:
248:
246:
245:
240:
238:
237:
215:Lie algebra case
190:
188:
187:
182:
170:
168:
167:
162:
150:
148:
147:
142:
140:
139:
123:
121:
120:
115:
110:
109:
81:
79:
78:
73:
54:
52:
51:
46:
44:
43:
1970:
1969:
1965:
1964:
1963:
1961:
1960:
1959:
1930:
1929:
1923:
1903:
1898:
1885:
1863:
1833:Fulton, William
1831:
1818:
1795:
1792:
1787:
1786:
1778:
1774:
1770:Corollary 13.20
1766:
1762:
1754:
1750:
1746:Definition 12.4
1742:
1738:
1730:
1726:
1718:
1714:
1699:10.2307/3647845
1680:
1679:
1675:
1667:
1660:
1652:
1645:
1637:
1633:
1628:
1606:
1562:
1538:
1537:
1515:
1514:
1495:
1494:
1471:
1470:
1451:
1450:
1431:
1430:
1411:
1410:
1374:
1369:
1368:
1331:
1330:
1294:
1293:
1274:
1273:
1250:
1249:
1203:
1202:
1179:
1178:
1159:
1158:
1135:
1134:
1111:
1110:
1064:
1063:
1040:
1039:
1034:be a connected
1016:
1015:
1012:
1006:
978:
977:
958:
957:
932:
931:
908:
907:
888:
887:
861:
860:
841:
840:
821:
820:
793:
792:
773:
772:
753:
752:
721:
720:
701:
700:
677:
676:
651:
646:
645:
626:
625:
606:
605:
577:
566:
565:
544:
533:
532:
495:
494:
465:
454:
453:
432:
427:
426:
407:
406:
373:
356:
346:
345:
313:
302:
301:
278:
277:
254:
253:
227:
226:
223:
217:
212:
173:
172:
153:
152:
129:
128:
101:
87:
86:
64:
63:
33:
32:
24:classifies the
12:
11:
5:
1968:
1966:
1958:
1957:
1952:
1947:
1942:
1932:
1931:
1928:
1927:
1921:
1915:, Birkhäuser,
1901:
1897:978-3319134666
1896:
1883:
1861:
1829:
1816:
1791:
1788:
1785:
1784:
1772:
1760:
1748:
1736:
1724:
1712:
1693:(5): 446–455.
1673:
1658:
1643:
1630:
1629:
1627:
1624:
1623:
1622:
1617:
1612:
1605:
1602:
1601:
1600:
1592:
1584:
1579:The theory of
1577:
1561:
1558:
1545:
1522:
1502:
1480:
1458:
1438:
1418:
1407:
1406:
1395:
1392:
1387:
1384:
1381:
1377:
1362:
1361:
1350:
1347:
1344:
1341:
1338:
1309:
1304:
1301:
1281:
1259:
1235:
1230:
1227:
1222:
1217:
1212:
1188:
1166:
1142:
1120:
1096:
1091:
1088:
1083:
1078:
1073:
1049:
1023:
1005:
1002:
998:
997:
985:
965:
954:
951:
939:
917:
895:
868:
848:
828:
817:highest weight
802:
780:
760:
734:
731:
728:
708:
684:
675:, we say that
662:
656:
633:
613:
588:
582:
576:
573:
551:
547:
543:
540:
520:
517:
514:
511:
508:
505:
502:
476:
470:
464:
461:
439:
435:
414:
403:
402:
388:
385:
382:
379:
376:
371:
368:
365:
362:
359:
353:
324:
318:
312:
309:
285:
263:
236:
216:
213:
211:
208:
180:
160:
138:
125:
124:
113:
108:
104:
100:
97:
94:
71:
42:
13:
10:
9:
6:
4:
3:
2:
1967:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1937:
1935:
1924:
1918:
1913:
1912:
1906:
1902:
1899:
1893:
1889:
1884:
1880:
1876:
1872:
1868:
1864:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1827:
1823:
1819:
1813:
1809:
1805:
1801:
1800:
1794:
1793:
1789:
1781:
1776:
1773:
1769:
1764:
1761:
1757:
1752:
1749:
1745:
1740:
1737:
1733:
1728:
1725:
1721:
1716:
1713:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1677:
1674:
1670:
1665:
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1659:
1655:
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1648:
1644:
1640:
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1632:
1625:
1621:
1618:
1616:
1613:
1611:
1608:
1607:
1603:
1597:
1593:
1589:
1585:
1582:
1581:Verma modules
1578:
1575:
1571:
1567:
1566:
1565:
1559:
1557:
1543:
1534:
1520:
1500:
1456:
1436:
1416:
1393:
1390:
1385:
1382:
1379:
1375:
1367:
1366:
1365:
1345:
1342:
1339:
1329:
1328:
1327:
1325:
1302:
1299:
1279:
1228:
1225:
1215:
1164:
1156:
1155:maximal torus
1140:
1089:
1086:
1076:
1037:
1021:
1011:
1003:
1001:
983:
963:
955:
952:
937:
893:
885:
884:
883:
880:
866:
846:
826:
818:
778:
758:
751:
746:
732:
729:
726:
706:
698:
682:
660:
631:
611:
603:
586:
574:
571:
564:. An element
549:
545:
541:
538:
518:
515:
509:
506:
503:
492:
474:
462:
459:
437:
433:
412:
383:
380:
377:
366:
363:
360:
351:
344:
343:
342:
340:
322:
310:
307:
299:
283:
252:
222:
214:
209:
207:
205:
201:
197:
192:
178:
158:
106:
102:
92:
85:
84:
83:
69:
62:
58:
31:
28:of a complex
27:
23:
19:
1945:Lie algebras
1910:
1887:
1840:
1798:
1775:
1763:
1751:
1739:
1727:
1715:
1690:
1686:
1676:
1671:Theorem 12.6
1639:Dixmier 1996
1634:
1563:
1535:
1408:
1363:
1323:
1013:
999:
881:
816:
747:
696:
601:
490:
404:
338:
224:
200:Hermann Weyl
193:
126:
21:
15:
1837:Harris, Joe
1734:Section 8.8
1722:Section 8.7
298:root system
196:Élie Cartan
1934:Categories
1790:References
1782:Chapter 12
1008:See also:
219:See also:
1907:(1972a),
1879:246650103
1780:Hall 2015
1768:Hall 2015
1756:Hall 2015
1744:Hall 2015
1732:Hall 2015
1720:Hall 2015
1669:Hall 2015
1654:Hall 2015
1383:π
1349:⟩
1340:λ
1337:⟨
1303:∈
1300:λ
984:λ
964:λ
847:μ
827:λ
759:λ
733:μ
730:−
727:λ
707:μ
683:λ
661:∗
632:μ
612:λ
587:∗
575:∈
572:λ
542:∈
539:α
516:≥
513:⟩
510:α
504:λ
501:⟨
475:∗
463:∈
460:λ
413:α
387:⟩
384:α
378:α
375:⟨
370:⟩
367:α
361:λ
358:⟨
323:∗
311:∈
308:λ
210:Statement
107:λ
96:↦
93:λ
61:Lie group
1839:(1991).
1604:See also
1572:and the
1062:and let
531:for all
491:dominant
339:integral
1871:1153249
1826:0498740
1707:3647845
1599:groups.
1591:proof.)
1201:. Then
930:, then
644:are in
1919:
1894:
1877:
1869:
1859:
1824:
1814:
1705:
1560:Proofs
1409:where
1133:. Let
750:weight
697:higher
276:. Let
1703:JSTOR
1626:Notes
1153:be a
699:than
1917:ISBN
1892:ISBN
1875:OCLC
1857:ISBN
1812:ISBN
1594:The
1586:The
1326:if
1014:Let
624:and
341:if
225:Let
1849:doi
1695:doi
1691:110
1322:is
1157:in
886:If
859:of
819:if
791:of
719:if
695:is
493:if
489:is
337:is
151:or
16:In
1936::
1873:.
1867:MR
1865:.
1855:.
1843:.
1835:;
1822:MR
1820:,
1810:,
1802:,
1701:.
1689:.
1685:.
1661:^
1646:^
1216::=
1077::=
879:.
748:A
206:.
1926:.
1881:.
1851::
1709:.
1697::
1576:.
1544:K
1521:K
1501:K
1479:g
1457:K
1437:K
1417:I
1394:I
1391:=
1386:H
1380:2
1376:e
1346:H
1343:,
1308:h
1280:R
1258:g
1234:t
1229:i
1226:+
1221:t
1211:h
1187:t
1165:K
1141:T
1119:g
1095:k
1090:i
1087:+
1082:k
1072:g
1048:k
1022:K
996:.
938:V
916:g
894:V
867:V
801:g
779:V
655:h
581:h
550:+
546:R
519:0
507:,
469:h
438:+
434:R
381:,
364:,
352:2
317:h
284:R
262:h
235:g
179:K
159:K
137:g
112:]
103:V
99:[
70:K
41:g
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