693:
1613:
35:
3313:, etc.). For each of these ways of defining Weyl groups, it is a (usually nontrivial) theorem that it is a Weyl group in the sense of the definition at the top of this article, namely the Weyl group of some root system associated with the object. A concrete realization of such a Weyl group usually depends on a choice – e.g. of
1949:
The figure illustrates the case of the A2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.
5054:
934:
3562:
3734:
3033:
The preceding claim is not hard to verify, if we simply remember what the Dynkin diagram tells us about the angle between each pair of roots. If, for example, there is no bond between the two vertices, then
3301:
Above, the Weyl group was defined as a subgroup of the isometry group of a root system. There are also various definitions of Weyl groups specific to various group-theoretic and geometric contexts (
1185:
root system, for example, the hyperplanes perpendicular to the roots are just lines, and the Weyl group is the symmetry group of an equilateral triangle, as indicated in the figure. As a group,
5665:
2259:
3228:
they are unconnected, connected by a simple edge, connected by a double edge, or connected by a triple edge. We have already noted these relations in the bullet points above, but to say that
2299:
1924:
3027:
2923:
2819:
2715:
1660:
2457:
4770:
1401:
5112:
4805:. Thus algebraic properties of the Weyl group correspond to general topological properties of manifolds. For instance, Poincaré duality gives a pairing between cells in dimension
4457:
1881:
1483:
1348:
3165:
Weyl groups are examples of finite reflection groups, as they are generated by reflections; the abstract groups (not considered as subgroups of a linear group) are accordingly
4064:
1855:
795:
970:
4821:
is the dimension of a manifold): the bottom (0) dimensional cell corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to the
4540:
1689:
4970:
4091:
3938:
2632:
2578:
2172:
1783:
1736:
1113:
1086:
822:
5242:
2659:
2605:
2064:
4269:
4186:
3844:
3115:
3135:
3092:
3052:
2971:
2867:
2763:
2551:
2502:
2411:
2367:
2327:
2212:
2192:
2125:
1709:
842:
4223:
3883:
3155:
3072:
2347:
2015:
1817:
4893:
4289:
3994:
3962:
2391:
2145:
1944:
1570:
1428:
1293:
1243:
1183:
1133:
1030:
745:
721:
4120:
3639:
3464:
1059:
1596:
4140:
4038:
4018:
3798:
3778:
3758:
3610:
3590:
3435:
3415:
3391:
3367:
3347:
3270:
3246:
2946:
2842:
2738:
2526:
2477:
2084:
2035:
1989:
1756:
1543:
1523:
1503:
1448:
1313:
1263:
1223:
1203:
1153:
1010:
990:
769:
4978:
5171:
is connected and either compact, or an affine algebraic group. The definition is simpler for a semisimple (or more generally reductive) Lie group over an
850:
4801:
of the group is related geometrically to the cohomology of the manifold (rather, of the real and complex forms of the group), which is constrained by
452:
3074:
are orthogonal, from which it follows easily that the corresponding reflections commute. More generally, the number of bonds determines the angle
3289:
of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. There is a unique
1960:: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.
500:
3472:
3647:
1205:
is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle. Note that in this case,
505:
495:
490:
4303:
as the reflection group generated by reflections in the roots – the specific realization of the root system depending on a choice of
5138:
5128:
4304:
681:
310:
1946:, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.
4895:; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the
574:
457:
5114:); for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.
5858:
5832:
5749:
5729:
5701:
5522:
5487:
5429:
5382:
605:
5586:
4822:
3290:
2221:
5882:
Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups",
5973:
5795:
2264:
1889:
4616:
2976:
2872:
2768:
2664:
1619:
467:
2416:
3997:
3285:
5928:
5514:
5460:
462:
442:
4727:
1353:
407:
315:
5074:
4389:
5923:
5455:
4142:
has the form of reflection. With a bit more effort, one can show that these reflections generate all of
1860:
1453:
1318:
5172:
3170:
1785:'s. The complement of the set of hyperplanes is disconnected, and each connected component is called a
1738:
denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of
447:
5870:(1984), "Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension",
4043:
1822:
774:
943:
4496:
1668:
646:
of that root system. Specifically, it is the subgroup which is generated by reflections through the
5918:
5556:
4924:
1607:
598:
82:
5957:
Jenn software for visualizing the Cayley graphs of finite
Coxeter groups on up to four generators
4930:
4069:
3891:
3174:
2610:
2556:
2150:
1761:
1714:
1091:
1064:
800:
654:
5450:
5205:
4853:!, and the number of elements of the general linear group over a finite field is related to the
2637:
2583:
2040:
5551:
4896:
4840:
4228:
4145:
677:
402:
365:
333:
320:
4651:. In general this is not always the case – the quotient does not always split, the normalizer
3806:
3097:
5978:
5822:
4899:, which considers Weyl groups to be simple algebraic groups over the field with one element.
3120:
3077:
3037:
2396:
2352:
2312:
2197:
2177:
2110:
1694:
827:
434:
102:
4191:
3853:
3140:
3057:
2332:
1994:
1796:
177:
167:
157:
147:
4862:
4719:
4679:
4620:
4274:
3967:
3947:
3166:
2376:
2130:
1929:
1548:
1406:
1271:
1228:
1161:
1118:
1015:
730:
699:
669:
662:
62:
52:
5903:
5842:
5814:
5787:
5759:
5711:
5532:
5497:
5439:
5411:
4096:
3615:
3440:
2504:
are the corresponding vertices in the Dynkin diagram. Then we have the following results:
1035:
8:
5983:
5392:
4471:
1575:
591:
579:
420:
250:
4802:
4562:
2951:
2847:
2743:
2531:
2482:
5569:
5424:, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press,
5049:{\displaystyle \operatorname {Out} (N)\cong H^{1}(W;T)\rtimes \operatorname {Out} (G).}
4656:
4644:
4125:
4023:
4003:
3783:
3763:
3743:
3595:
3575:
3420:
3400:
3376:
3352:
3332:
3255:
3231:
2931:
2827:
2723:
2511:
2462:
2069:
2020:
1974:
1741:
1528:
1508:
1488:
1433:
1298:
1248:
1208:
1188:
1138:
995:
975:
754:
351:
341:
4647:
of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup of
5937:
5854:
5828:
5745:
5725:
5697:
5518:
5483:
5446:
5425:
5378:
5123:
3314:
415:
378:
5504:
3296:
530:
268:
5899:
5891:
5838:
5810:
5783:
5775:
5755:
5707:
5676:
5580:
5561:
5539:
5528:
5493:
5435:
5407:
4912:
4623:, but with any non-zero numbers in place of the '1's), and whose Weyl group is the
550:
230:
222:
214:
206:
198:
131:
112:
72:
5940:
5393:"Automorphisms of Normalizers of Maximal Tori and First Cohomology of Weyl Groups"
3094:
between the roots. The product of the two reflections is then a rotation by angle
1225:
is not the full symmetry group of the root system; a 60-degree rotation preserves
5796:"On the second cohomology groups (Schur-multipliers) of finite reflection groups"
5739:
5719:
5508:
5477:
5419:
5372:
5180:
4849:
and Weyl groups – for instance, the number of elements of the symmetric group is
4846:
4697:
4624:
3310:
3280:
929:{\displaystyle s_{\alpha }(v)=v-2{\frac {(v,\alpha )}{(\alpha ,\alpha )}}\alpha }
535:
288:
273:
44:
5688:
5779:
5680:
5064:
4700:
4689:
3225:
2370:
643:
555:
373:
278:
540:
5967:
5867:
5143:
5133:
4798:
4490:
3370:
3318:
263:
92:
1691:
is a root system, we may consider the hyperplane perpendicular to each root
4791:
4776:
4581:, but the resulting groups are all isomorphic (by an inner automorphism of
3276:
631:
560:
545:
346:
328:
258:
692:
5766:
Howlett, Robert B. (1988), "On the Schur
Multipliers of Coxeter Groups",
5377:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
5374:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
4854:
4357:
3885:, at which point one has an alternative description of the Weyl group as
3302:
1789:. If we have fixed a particular set Δ of simple roots, we may define the
748:
673:
635:
623:
619:
386:
302:
26:
4604:
is isomorphic to the Weyl group of its Lie algebra, as discussed above.
1350:
whose entries sum to zero. The roots consist of the vectors of the form
5573:
4330:
3557:{\displaystyle N(T)=\{x\in K|xtx^{-1}\in T,\,{\text{for all }}t\in T\}}
3157:, as the reader may verify, from which the above claim follows easily.
1612:
650:
647:
525:
283:
5853:, Progress in Mathematics, vol. 140 (2nd ed.), Birkhaeuser,
4577:). Note that the specific quotient set depends on a choice of maximal
3173:. Being a Coxeter group means that a Weyl group has a special kind of
661:
finite reflection groups are Weyl groups. Abstractly, Weyl groups are
5945:
5895:
4311:
3306:
22:
5565:
3224:
make an angle of 90, 120, 135, or 150 degrees, i.e., whether in the
3209:)=1. The generators are the reflections given by simple roots, and
1485:. The reflection associated to such a root is the transformation of
5955:
4834:
3729:{\displaystyle Z(T)=\{x\in K|xtx^{-1}=t\,{\text{for all }}t\in T\}}
639:
3324:
3297:
Weyl groups in algebraic, group-theoretic, and geometric settings
482:
5583:(1935), "The complete enumeration of finite groups of the form
4299:
For a complex semisimple Lie algebra, the Weyl group is simply
1616:
The shaded region is the fundamental Weyl chamber for the base
4667:
and the Weyl group cannot always be realized as a subgroup of
4578:
824:
denote the reflection about the hyperplane perpendicular to
723:
root system is the symmetry group of an equilateral triangle
34:
5129:
Semisimple Lie algebra#Cartan subalgebras and root systems
5724:, Graduate texts in mathematics, vol. 99, Springer,
4639:
splits (via the permutation matrices), so the normalizer
4615:
of invertible diagonal matrices, whose normalizer is the
3293:, which is opposite to the identity in the Bruhat order.
5744:, Research Notes in Mathematics, vol. 54, Pitman,
2661:
each have order two, this is equivalent to saying that
2218:
That is to say, the group generated by the reflections
5935:
2261:
is the same as the group generated by the reflections
2147:, then the Weyl group is generated by the reflections
5589:
5257:
5208:
5167:
Different conditions are sufficient – most simply if
5077:
4981:
4933:
4865:
4730:
4499:
4392:
4277:
4231:
4194:
4148:
4128:
4099:
4072:
4046:
4026:
4006:
3970:
3950:
3894:
3856:
3809:
3786:
3766:
3746:
3650:
3618:
3598:
3578:
3475:
3443:
3423:
3403:
3379:
3355:
3335:
3258:
3248:
is a
Coxeter group, we are saying that those are the
3234:
3143:
3123:
3100:
3080:
3060:
3040:
2979:
2954:
2934:
2875:
2850:
2830:
2771:
2746:
2726:
2667:
2640:
2613:
2586:
2559:
2534:
2514:
2485:
2465:
2419:
2399:
2379:
2355:
2335:
2315:
2267:
2224:
2200:
2180:
2153:
2133:
2113:
2072:
2043:
2023:
1997:
1977:
1953:
A basic general theorem about Weyl chambers is this:
1932:
1892:
1863:
1825:
1799:
1764:
1744:
1717:
1697:
1671:
1622:
1578:
1551:
1531:
1511:
1491:
1456:
1436:
1409:
1356:
1321:
1301:
1274:
1251:
1231:
1211:
1191:
1164:
1141:
1121:
1094:
1067:
1038:
1018:
998:
978:
946:
853:
830:
803:
777:
757:
733:
702:
5542:(1934), "Discrete groups generated by reflections",
5390:
5355:
5068:
5063:) are essentially the diagram automorphisms of the
4271:is isomorphic to the Weyl group of the root system
5659:
5236:
5106:
5048:
4964:
4887:
4764:
4534:
4451:
4283:
4263:
4217:
4180:
4134:
4114:
4085:
4058:
4032:
4012:
3988:
3956:
3932:
3877:
3838:
3792:
3772:
3752:
3728:
3633:
3604:
3584:
3556:
3458:
3429:
3409:
3385:
3361:
3341:
3264:
3240:
3149:
3129:
3109:
3086:
3066:
3046:
3021:
2965:
2940:
2917:
2861:
2836:
2813:
2757:
2732:
2709:
2653:
2626:
2599:
2572:
2545:
2520:
2496:
2471:
2451:
2405:
2385:
2361:
2341:
2321:
2293:
2253:
2206:
2186:
2166:
2139:
2119:
2078:
2058:
2029:
2009:
1983:
1938:
1918:
1875:
1849:
1811:
1777:
1750:
1730:
1703:
1683:
1654:
1590:
1564:
1537:
1517:
1497:
1477:
1442:
1422:
1395:
1342:
1307:
1287:
1257:
1237:
1217:
1197:
1177:
1147:
1127:
1107:
1080:
1053:
1024:
1004:
984:
964:
928:
836:
816:
789:
763:
739:
715:
5660:{\displaystyle r_{i}^{2}=(r_{i}r_{j})^{k_{ij}}=1}
4188:. Thus, in the end, the Weyl group as defined as
2254:{\displaystyle s_{\alpha },\,\alpha \in \Delta ,}
5965:
5391:Hämmerli, J.-F.; Matthey, M.; Suter, U. (2004),
4828:
453:Representation theory of semisimple Lie algebras
3325:The Weyl group of a connected compact Lie group
4907:For a non-abelian connected compact Lie group
3184:is of order two, and the relations other than
3169:, which allows them to be classified by their
2294:{\displaystyle s_{\alpha },\,\alpha \in \Phi }
1919:{\displaystyle s_{\alpha },\,\alpha \in \Phi }
1545:th entries of each vector. The Weyl group for
5793:
5503:
4775:which gives rise to the decomposition of the
4329:that torus is defined as the quotient of the
4317:satisfying certain conditions, given a torus
3022:{\displaystyle (s_{\alpha }s_{\beta })^{6}=1}
2918:{\displaystyle (s_{\alpha }s_{\beta })^{4}=1}
2814:{\displaystyle (s_{\alpha }s_{\beta })^{3}=1}
2710:{\displaystyle (s_{\alpha }s_{\beta })^{2}=1}
1088:'s. By the definition of a root system, each
599:
5445:
5314:
5202:– that is how it is defined – and the group
5071:and is a finite elementary abelian 2-group (
5067:, while the group cohomology is computed in
4325:(which need not be maximal), the Weyl group
3723:
3666:
3551:
3491:
3216:is 2, 3, 4, or 6 depending on whether roots
2446:
2420:
1649:
1623:
5717:
5690:The Geometry and Topology of Coxeter Groups
5324:
5322:
2100:A key result about the Weyl group is this:
1655:{\displaystyle \{\alpha _{1},\alpha _{2}\}}
5718:Grove, Larry C.; Benson, Clark T. (1985),
5310:
5308:
4607:For example, for the general linear group
2452:{\displaystyle \{s_{\alpha },s_{\beta }\}}
2090:
2037:contains exactly one point in the closure
1608:Coxeter group § Affine Coxeter groups
687:
606:
592:
491:Particle physics and representation theory
33:
16:Subgroup of a root system's isometry group
5555:
5479:Lie Groups and Lie Algebras: Chapters 4-6
5417:
5263:
5059:The outer automorphisms of the group Out(
3708:
3536:
3349:be a connected compact Lie group and let
2281:
2238:
1906:
1459:
1383:
1324:
5881:
5475:
5319:
5139:Root system of a semi-simple Lie algebra
4845:There are a number of analogies between
1611:
1032:is the subgroup of the orthogonal group
691:
5866:
5765:
5579:
5538:
5305:
4919:with coefficients in the maximal torus
4673:
665:, and are important examples of these.
458:Representations of classical Lie groups
5966:
5827:, CMS Books in Mathematics, Springer,
5824:Reflection Groups and Invariant Theory
5737:
5351:
5349:
4765:{\displaystyle G=\bigcup _{w\in W}BwB}
2413:tells us something about how the pair
5936:
5848:
5686:
5482:, Elements of Mathematics, Springer,
4923:used to define it, is related to the
4585:), since maximal tori are conjugate.
3780:(relative to the given maximal torus
3160:
1793:associated to Δ as the set of points
1396:{\displaystyle e_{i}-e_{j},\,i\neq j}
5820:
5421:Reflection Groups and Coxeter Groups
5370:
5340:
5328:
5299:
5287:
5275:
5244:means "with respect to this action".
5107:{\displaystyle (\mathbf {Z} /2)^{k}}
4452:{\displaystyle W(T,G):=N(T)/Z(T).\ }
4294:
682:root system of that group or algebra
311:Lie group–Lie algebra correspondence
5794:Ihara, S.; Yokonuma, Takeo (1965),
5346:
4493:(so it equals its own centralizer:
3283:in terms of this presentation: the
13:
5851:Lie Groups: Beyond an Introduction
5468:
5356:Hämmerli, Matthey & Suter 2004
5069:Hämmerli, Matthey & Suter 2004
4823:longest element of a Coxeter group
4278:
4053:
3951:
3944:Now, one can define a root system
3291:longest element of a Coxeter group
2400:
2380:
2356:
2288:
2245:
2201:
2134:
2114:
1933:
1913:
1876:{\displaystyle \alpha \in \Delta }
1870:
1672:
1478:{\displaystyle \mathbb {R} ^{n+1}}
1343:{\displaystyle \mathbb {R} ^{n+1}}
1232:
1122:
1019:
784:
734:
14:
5995:
5911:
5884:J. Fac. Sci. Univ. Tokyo, Sect. 1
5803:J. Fac. Sci. Univ. Tokyo, Sect. 1
2928:If there are three bonds between
2095:
1572:is then the permutation group on
5179:Weyl group can be defined for a
5082:
4627:. In this case the quotient map
4617:generalized permutation matrices
4611:a maximal torus is the subgroup
4059:{\displaystyle \alpha \in \Phi }
1850:{\displaystyle (\alpha ,v)>0}
1601:
790:{\displaystyle \alpha \in \Phi }
680:, etc. is the Weyl group of the
5510:Combinatorics of Coxeter Groups
4066:, one can construct an element
3800:) is then defined initially as
2824:If there are two bonds between
2459:behaves. Specifically, suppose
1315:is the space of all vectors in
965:{\displaystyle (\cdot ,\cdot )}
844:, which is given explicitly as
653:to the roots, and as such is a
5696:, Princeton University Press,
5632:
5608:
5406:, Heldermann Verlag: 583–617,
5334:
5293:
5281:
5269:
5231:
5219:
5189:
5161:
5095:
5078:
5040:
5034:
5022:
5010:
4994:
4988:
4956:
4950:
4873:
4866:
4600:torus, then the Weyl group of
4592:is compact and connected, and
4542:) then the resulting quotient
4535:{\displaystyle Z(T_{0})=T_{0}}
4516:
4503:
4440:
4434:
4423:
4417:
4408:
4396:
4258:
4252:
4241:
4235:
4204:
4198:
4175:
4169:
4158:
4152:
4109:
4103:
3983:
3971:
3927:
3921:
3910:
3904:
3866:
3860:
3825:
3819:
3679:
3660:
3654:
3628:
3622:
3504:
3485:
3479:
3453:
3447:
3004:
2980:
2900:
2876:
2796:
2772:
2692:
2668:
2050:
1964:A related result is this one:
1838:
1826:
1684:{\displaystyle \Phi \subset V}
1505:obtained by interchanging the
1450:th standard basis element for
1048:
1042:
959:
947:
917:
905:
900:
888:
870:
864:
622:, in particular the theory of
506:Galilean group representations
501:Poincaré group representations
1:
5515:Graduate Texts in Mathematics
5418:Humphreys, James E. (1992) ,
5363:
4902:
4829:Analogy with algebraic groups
4703:subgroup and a maximal torus
2720:If there is one bond between
1135:, from which it follows that
496:Lorentz group representations
463:Theorem of the highest weight
5507:; Brenti, Francesco (2005),
5251:
5149:
3996:; the roots are the nonzero
3850:Eventually, one proves that
2508:If there is no bond between
2304:
657:. In fact it turns out that
7:
5924:Encyclopedia of Mathematics
5517:, vol. 231, Springer,
5456:Encyclopedia of Mathematics
5117:
4965:{\displaystyle N=N_{G}(T),}
4086:{\displaystyle x_{\alpha }}
3933:{\displaystyle W=N(T)/Z(T)}
2627:{\displaystyle s_{\alpha }}
2573:{\displaystyle s_{\alpha }}
2167:{\displaystyle s_{\alpha }}
1778:{\displaystyle s_{\alpha }}
1731:{\displaystyle s_{\alpha }}
1295:root system. In this case,
1108:{\displaystyle s_{\alpha }}
1081:{\displaystyle s_{\alpha }}
817:{\displaystyle s_{\alpha }}
10:
6000:
5872:Trudy Moskov. Mat. Obshch.
5849:Knapp, Anthony W. (2002),
5741:Geometry of Coxeter groups
5687:Davis, Michael W. (2007),
5476:Bourbaki, Nicolas (2002),
5278:Propositions 8.23 and 8.27
5237:{\displaystyle H^{1}(W;T)}
5173:algebraically closed field
4838:
4832:
4677:
2654:{\displaystyle s_{\beta }}
2600:{\displaystyle s_{\beta }}
2059:{\displaystyle {\bar {C}}}
1605:
448:Lie algebra representation
5449:; Fedenko, A.S. (2001) ,
4619:(matrices in the form of
4264:{\displaystyle N(T)/Z(T)}
4181:{\displaystyle N(T)/Z(T)}
4000:of the adjoint action of
1268:We may consider also the
1245:but is not an element of
5974:Finite reflection groups
5780:10.1112/jlms/s2-38.2.263
5721:Finite Reflection Groups
5681:10.1112/jlms/s1-10.37.21
5315:Popov & Fedenko 2001
5154:
4925:outer automorphism group
3839:{\displaystyle W=N(T)/T}
3393:. We then introduce the
3177:in which each generator
3117:in the plane spanned by
3110:{\displaystyle 2\theta }
1791:fundamental Weyl chamber
972:is the inner product on
443:Lie group representation
5738:Hiller, Howard (1982),
5371:Hall, Brian C. (2015),
3964:associated to the pair
3130:{\displaystyle \alpha }
3087:{\displaystyle \theta }
3047:{\displaystyle \alpha }
2406:{\displaystyle \Delta }
2362:{\displaystyle \Delta }
2322:{\displaystyle \alpha }
2207:{\displaystyle \Delta }
2187:{\displaystyle \alpha }
2120:{\displaystyle \Delta }
2091:Coxeter group structure
1704:{\displaystyle \alpha }
837:{\displaystyle \alpha }
688:Definition and examples
655:finite reflection group
468:Borel–Weil–Bott theorem
5821:Kane, Richard (2001),
5661:
5238:
5108:
5050:
4966:
4897:field with one element
4889:
4841:Field with one element
4766:
4536:
4453:
4285:
4265:
4219:
4218:{\displaystyle N(T)/T}
4182:
4136:
4116:
4087:
4060:
4034:
4020:on the Lie algebra of
4014:
3990:
3958:
3934:
3879:
3878:{\displaystyle Z(T)=T}
3840:
3794:
3774:
3754:
3730:
3635:
3606:
3586:
3558:
3460:
3431:
3411:
3387:
3363:
3343:
3317:for a Lie algebra, of
3266:
3242:
3171:Coxeter–Dynkin diagram
3151:
3150:{\displaystyle \beta }
3131:
3111:
3088:
3068:
3067:{\displaystyle \beta }
3048:
3023:
2967:
2942:
2919:
2863:
2838:
2815:
2759:
2734:
2711:
2655:
2628:
2601:
2574:
2547:
2522:
2498:
2473:
2453:
2407:
2387:
2363:
2343:
2342:{\displaystyle \beta }
2323:
2295:
2255:
2208:
2188:
2168:
2141:
2121:
2080:
2060:
2031:
2011:
2010:{\displaystyle v\in V}
1985:
1940:
1920:
1886:Since the reflections
1877:
1851:
1813:
1812:{\displaystyle v\in V}
1779:
1752:
1732:
1705:
1685:
1662:
1656:
1592:
1566:
1539:
1519:
1499:
1479:
1444:
1424:
1397:
1344:
1309:
1289:
1259:
1239:
1219:
1199:
1179:
1149:
1129:
1109:
1082:
1055:
1026:
1006:
986:
966:
930:
838:
818:
791:
765:
741:
724:
717:
696:The Weyl group of the
678:linear algebraic group
366:Semisimple Lie algebra
321:Adjoint representation
5662:
5400:Journal of Lie Theory
5239:
5109:
5051:
4967:
4890:
4888:{\displaystyle _{q}!}
4797:The structure of the
4767:
4718:, then we obtain the
4678:Further information:
4537:
4454:
4286:
4284:{\displaystyle \Phi }
4266:
4220:
4183:
4137:
4117:
4088:
4061:
4035:
4015:
3991:
3989:{\displaystyle (K,T)}
3959:
3957:{\displaystyle \Phi }
3935:
3880:
3841:
3795:
3775:
3755:
3731:
3636:
3607:
3587:
3559:
3461:
3432:
3412:
3388:
3364:
3344:
3267:
3243:
3167:finite Coxeter groups
3152:
3132:
3112:
3089:
3069:
3049:
3024:
2968:
2943:
2920:
2864:
2839:
2816:
2760:
2735:
2712:
2656:
2629:
2602:
2575:
2548:
2523:
2499:
2474:
2454:
2408:
2393:relative to the base
2388:
2386:{\displaystyle \Phi }
2364:
2344:
2324:
2296:
2256:
2209:
2189:
2169:
2142:
2140:{\displaystyle \Phi }
2122:
2081:
2061:
2032:
2012:
1986:
1971:: Fix a Weyl chamber
1941:
1939:{\displaystyle \Phi }
1921:
1878:
1852:
1814:
1780:
1758:generated by all the
1753:
1733:
1706:
1686:
1657:
1615:
1593:
1567:
1565:{\displaystyle A_{n}}
1540:
1520:
1500:
1480:
1445:
1425:
1423:{\displaystyle e_{i}}
1398:
1345:
1310:
1290:
1288:{\displaystyle A_{n}}
1260:
1240:
1238:{\displaystyle \Phi }
1220:
1200:
1180:
1178:{\displaystyle A_{2}}
1150:
1130:
1128:{\displaystyle \Phi }
1110:
1083:
1061:generated by all the
1056:
1027:
1025:{\displaystyle \Phi }
1007:
987:
967:
931:
839:
819:
792:
766:
751:in a Euclidean space
742:
740:{\displaystyle \Phi }
718:
716:{\displaystyle A_{2}}
695:
663:finite Coxeter groups
435:Representation theory
5768:J. London Math. Soc.
5669:J. London Math. Soc.
5587:
5206:
5075:
4979:
4931:
4863:
4728:
4720:Bruhat decomposition
4714:is chosen to lie in
4680:Bruhat decomposition
4674:Bruhat decomposition
4621:permutation matrices
4497:
4390:
4275:
4229:
4192:
4146:
4126:
4115:{\displaystyle N(T)}
4097:
4070:
4044:
4024:
4004:
3968:
3948:
3892:
3854:
3807:
3784:
3764:
3744:
3648:
3634:{\displaystyle Z(T)}
3616:
3596:
3576:
3473:
3459:{\displaystyle N(T)}
3441:
3421:
3401:
3377:
3353:
3333:
3256:
3232:
3141:
3121:
3098:
3078:
3058:
3038:
2977:
2952:
2932:
2873:
2848:
2828:
2769:
2744:
2724:
2665:
2638:
2611:
2584:
2557:
2532:
2512:
2483:
2463:
2417:
2397:
2377:
2353:
2333:
2313:
2265:
2222:
2198:
2178:
2151:
2131:
2111:
2070:
2041:
2021:
2017:, the Weyl-orbit of
1995:
1975:
1930:
1890:
1861:
1823:
1797:
1762:
1742:
1715:
1695:
1669:
1620:
1576:
1549:
1529:
1509:
1489:
1454:
1434:
1407:
1354:
1319:
1299:
1272:
1249:
1229:
1209:
1189:
1162:
1139:
1119:
1092:
1065:
1054:{\displaystyle O(V)}
1036:
1016:
996:
976:
944:
851:
828:
801:
775:
755:
731:
700:
670:semisimple Lie group
668:The Weyl group of a
5604:
3568:We also define the
3275:Weyl groups have a
1591:{\displaystyle n+1}
1158:In the case of the
1155:is a finite group.
580:Table of Lie groups
421:Compact Lie algebra
5938:Weisstein, Eric W.
5657:
5590:
5343:Propositions 11.35
5234:
5104:
5046:
4962:
4927:of the normalizer
4915:of the Weyl group
4885:
4762:
4752:
4696:, i.e., a maximal
4657:semidirect product
4655:is not always the
4645:semidirect product
4532:
4449:
4281:
4261:
4215:
4178:
4132:
4112:
4083:
4056:
4030:
4010:
3986:
3954:
3930:
3875:
3836:
3790:
3770:
3750:
3726:
3631:
3602:
3582:
3554:
3456:
3427:
3407:
3383:
3359:
3339:
3262:
3238:
3161:As a Coxeter group
3147:
3127:
3107:
3084:
3064:
3044:
3019:
2966:{\displaystyle v'}
2963:
2938:
2915:
2862:{\displaystyle v'}
2859:
2834:
2811:
2758:{\displaystyle v'}
2755:
2730:
2707:
2651:
2624:
2597:
2570:
2546:{\displaystyle v'}
2543:
2518:
2497:{\displaystyle v'}
2494:
2469:
2449:
2403:
2383:
2359:
2339:
2319:
2291:
2251:
2204:
2184:
2164:
2137:
2117:
2076:
2056:
2027:
2007:
1981:
1936:
1916:
1873:
1847:
1809:
1775:
1748:
1728:
1701:
1681:
1663:
1652:
1588:
1562:
1535:
1515:
1495:
1475:
1440:
1420:
1393:
1340:
1305:
1285:
1255:
1235:
1215:
1195:
1175:
1145:
1125:
1105:
1078:
1051:
1022:
1002:
982:
962:
926:
834:
814:
787:
761:
737:
725:
713:
352:Affine Lie algebra
342:Simple Lie algebra
83:Special orthogonal
5860:978-0-8176-4259-4
5834:978-0-387-98979-2
5751:978-0-273-08517-1
5731:978-0-387-96082-1
5703:978-0-691-13138-2
5581:Coxeter, H. S. M.
5540:Coxeter, H. S. M.
5524:978-3-540-27596-1
5489:978-3-540-42650-9
5431:978-0-521-43613-7
5384:978-3-319-13466-6
5302:Propositions 8.24
5124:Affine Weyl group
4809:and in dimension
4737:
4448:
4305:Cartan subalgebra
4295:In other settings
4135:{\displaystyle T}
4033:{\displaystyle K}
4013:{\displaystyle T}
3793:{\displaystyle T}
3773:{\displaystyle K}
3753:{\displaystyle W}
3712:
3605:{\displaystyle K}
3585:{\displaystyle T}
3540:
3430:{\displaystyle K}
3410:{\displaystyle T}
3386:{\displaystyle K}
3362:{\displaystyle T}
3342:{\displaystyle K}
3321:for a Lie group.
3315:Cartan subalgebra
3265:{\displaystyle W}
3241:{\displaystyle W}
3192:are of the form (
2941:{\displaystyle v}
2837:{\displaystyle v}
2733:{\displaystyle v}
2521:{\displaystyle v}
2472:{\displaystyle v}
2079:{\displaystyle C}
2053:
2030:{\displaystyle v}
1984:{\displaystyle C}
1751:{\displaystyle V}
1538:{\displaystyle j}
1518:{\displaystyle i}
1498:{\displaystyle V}
1443:{\displaystyle i}
1308:{\displaystyle V}
1258:{\displaystyle W}
1218:{\displaystyle W}
1198:{\displaystyle W}
1148:{\displaystyle W}
1005:{\displaystyle W}
992:. The Weyl group
985:{\displaystyle V}
921:
764:{\displaystyle V}
616:
615:
416:Split Lie algebra
379:Cartan subalgebra
241:
240:
132:Simple Lie groups
5991:
5960:
5951:
5950:
5932:
5906:
5878:
5863:
5845:
5817:
5800:
5790:
5762:
5734:
5714:
5695:
5683:
5666:
5664:
5663:
5658:
5650:
5649:
5648:
5647:
5630:
5629:
5620:
5619:
5603:
5598:
5576:
5559:
5535:
5500:
5463:
5442:
5414:
5397:
5387:
5358:
5353:
5344:
5338:
5332:
5326:
5317:
5312:
5303:
5297:
5291:
5290:Proposition 8.29
5285:
5279:
5273:
5267:
5261:
5245:
5243:
5241:
5240:
5235:
5218:
5217:
5193:
5187:
5165:
5113:
5111:
5110:
5105:
5103:
5102:
5090:
5085:
5055:
5053:
5052:
5047:
5009:
5008:
4971:
4969:
4968:
4963:
4949:
4948:
4913:group cohomology
4894:
4892:
4891:
4886:
4881:
4880:
4847:algebraic groups
4803:Poincaré duality
4771:
4769:
4768:
4763:
4751:
4541:
4539:
4538:
4533:
4531:
4530:
4515:
4514:
4458:
4456:
4455:
4450:
4446:
4430:
4290:
4288:
4287:
4282:
4270:
4268:
4267:
4262:
4248:
4224:
4222:
4221:
4216:
4211:
4187:
4185:
4184:
4179:
4165:
4141:
4139:
4138:
4133:
4122:whose action on
4121:
4119:
4118:
4113:
4092:
4090:
4089:
4084:
4082:
4081:
4065:
4063:
4062:
4057:
4039:
4037:
4036:
4031:
4019:
4017:
4016:
4011:
3995:
3993:
3992:
3987:
3963:
3961:
3960:
3955:
3939:
3937:
3936:
3931:
3917:
3884:
3882:
3881:
3876:
3845:
3843:
3842:
3837:
3832:
3799:
3797:
3796:
3791:
3779:
3777:
3776:
3771:
3759:
3757:
3756:
3751:
3735:
3733:
3732:
3727:
3713:
3710:
3701:
3700:
3682:
3640:
3638:
3637:
3632:
3611:
3609:
3608:
3603:
3591:
3589:
3588:
3583:
3563:
3561:
3560:
3555:
3541:
3538:
3526:
3525:
3507:
3465:
3463:
3462:
3457:
3436:
3434:
3433:
3428:
3416:
3414:
3413:
3408:
3392:
3390:
3389:
3384:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3271:
3269:
3268:
3263:
3247:
3245:
3244:
3239:
3156:
3154:
3153:
3148:
3136:
3134:
3133:
3128:
3116:
3114:
3113:
3108:
3093:
3091:
3090:
3085:
3073:
3071:
3070:
3065:
3053:
3051:
3050:
3045:
3028:
3026:
3025:
3020:
3012:
3011:
3002:
3001:
2992:
2991:
2972:
2970:
2969:
2964:
2962:
2947:
2945:
2944:
2939:
2924:
2922:
2921:
2916:
2908:
2907:
2898:
2897:
2888:
2887:
2868:
2866:
2865:
2860:
2858:
2843:
2841:
2840:
2835:
2820:
2818:
2817:
2812:
2804:
2803:
2794:
2793:
2784:
2783:
2764:
2762:
2761:
2756:
2754:
2739:
2737:
2736:
2731:
2716:
2714:
2713:
2708:
2700:
2699:
2690:
2689:
2680:
2679:
2660:
2658:
2657:
2652:
2650:
2649:
2633:
2631:
2630:
2625:
2623:
2622:
2606:
2604:
2603:
2598:
2596:
2595:
2579:
2577:
2576:
2571:
2569:
2568:
2552:
2550:
2549:
2544:
2542:
2527:
2525:
2524:
2519:
2503:
2501:
2500:
2495:
2493:
2478:
2476:
2475:
2470:
2458:
2456:
2455:
2450:
2445:
2444:
2432:
2431:
2412:
2410:
2409:
2404:
2392:
2390:
2389:
2384:
2368:
2366:
2365:
2360:
2348:
2346:
2345:
2340:
2328:
2326:
2325:
2320:
2300:
2298:
2297:
2292:
2277:
2276:
2260:
2258:
2257:
2252:
2234:
2233:
2213:
2211:
2210:
2205:
2193:
2191:
2190:
2185:
2173:
2171:
2170:
2165:
2163:
2162:
2146:
2144:
2143:
2138:
2126:
2124:
2123:
2118:
2085:
2083:
2082:
2077:
2065:
2063:
2062:
2057:
2055:
2054:
2046:
2036:
2034:
2033:
2028:
2016:
2014:
2013:
2008:
1990:
1988:
1987:
1982:
1945:
1943:
1942:
1937:
1925:
1923:
1922:
1917:
1902:
1901:
1882:
1880:
1879:
1874:
1856:
1854:
1853:
1848:
1818:
1816:
1815:
1810:
1784:
1782:
1781:
1776:
1774:
1773:
1757:
1755:
1754:
1749:
1737:
1735:
1734:
1729:
1727:
1726:
1710:
1708:
1707:
1702:
1690:
1688:
1687:
1682:
1661:
1659:
1658:
1653:
1648:
1647:
1635:
1634:
1597:
1595:
1594:
1589:
1571:
1569:
1568:
1563:
1561:
1560:
1544:
1542:
1541:
1536:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1484:
1482:
1481:
1476:
1474:
1473:
1462:
1449:
1447:
1446:
1441:
1429:
1427:
1426:
1421:
1419:
1418:
1402:
1400:
1399:
1394:
1379:
1378:
1366:
1365:
1349:
1347:
1346:
1341:
1339:
1338:
1327:
1314:
1312:
1311:
1306:
1294:
1292:
1291:
1286:
1284:
1283:
1264:
1262:
1261:
1256:
1244:
1242:
1241:
1236:
1224:
1222:
1221:
1216:
1204:
1202:
1201:
1196:
1184:
1182:
1181:
1176:
1174:
1173:
1154:
1152:
1151:
1146:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1104:
1103:
1087:
1085:
1084:
1079:
1077:
1076:
1060:
1058:
1057:
1052:
1031:
1029:
1028:
1023:
1011:
1009:
1008:
1003:
991:
989:
988:
983:
971:
969:
968:
963:
935:
933:
932:
927:
922:
920:
903:
886:
863:
862:
843:
841:
840:
835:
823:
821:
820:
815:
813:
812:
796:
794:
793:
788:
771:. For each root
770:
768:
767:
762:
746:
744:
743:
738:
722:
720:
719:
714:
712:
711:
608:
601:
594:
551:Claude Chevalley
408:Complexification
251:Other Lie groups
137:
136:
45:Classical groups
37:
19:
18:
5999:
5998:
5994:
5993:
5992:
5990:
5989:
5988:
5964:
5963:
5954:
5941:"Coxeter group"
5919:"Coxeter group"
5917:
5914:
5909:
5861:
5835:
5798:
5752:
5732:
5704:
5693:
5640:
5636:
5635:
5631:
5625:
5621:
5615:
5611:
5599:
5594:
5588:
5585:
5584:
5566:10.2307/1968753
5525:
5505:Björner, Anders
5490:
5471:
5469:Further reading
5466:
5432:
5395:
5385:
5366:
5361:
5354:
5347:
5339:
5335:
5327:
5320:
5313:
5306:
5298:
5294:
5286:
5282:
5274:
5270:
5262:
5258:
5254:
5249:
5248:
5213:
5209:
5207:
5204:
5203:
5194:
5190:
5166:
5162:
5157:
5152:
5120:
5098:
5094:
5086:
5081:
5076:
5073:
5072:
5004:
5000:
4980:
4977:
4976:
4944:
4940:
4932:
4929:
4928:
4905:
4876:
4872:
4864:
4861:
4860:
4843:
4837:
4831:
4741:
4729:
4726:
4725:
4713:
4682:
4676:
4625:symmetric group
4526:
4522:
4510:
4506:
4498:
4495:
4494:
4488:
4426:
4391:
4388:
4387:
4377:
4350:
4327:with respect to
4297:
4276:
4273:
4272:
4244:
4230:
4227:
4226:
4207:
4193:
4190:
4189:
4161:
4147:
4144:
4143:
4127:
4124:
4123:
4098:
4095:
4094:
4077:
4073:
4071:
4068:
4067:
4045:
4042:
4041:
4025:
4022:
4021:
4005:
4002:
4001:
3969:
3966:
3965:
3949:
3946:
3945:
3913:
3893:
3890:
3889:
3855:
3852:
3851:
3828:
3808:
3805:
3804:
3785:
3782:
3781:
3765:
3762:
3761:
3745:
3742:
3741:
3740:The Weyl group
3709:
3693:
3689:
3678:
3649:
3646:
3645:
3641:and defined as
3617:
3614:
3613:
3597:
3594:
3593:
3577:
3574:
3573:
3537:
3518:
3514:
3503:
3474:
3471:
3470:
3466:and defined as
3442:
3439:
3438:
3422:
3419:
3418:
3402:
3399:
3398:
3378:
3375:
3374:
3354:
3351:
3350:
3334:
3331:
3330:
3327:
3311:symmetric space
3299:
3281:length function
3257:
3254:
3253:
3233:
3230:
3229:
3214:
3208:
3200:
3189:
3182:
3163:
3142:
3139:
3138:
3122:
3119:
3118:
3099:
3096:
3095:
3079:
3076:
3075:
3059:
3056:
3055:
3039:
3036:
3035:
3007:
3003:
2997:
2993:
2987:
2983:
2978:
2975:
2974:
2955:
2953:
2950:
2949:
2933:
2930:
2929:
2903:
2899:
2893:
2889:
2883:
2879:
2874:
2871:
2870:
2851:
2849:
2846:
2845:
2829:
2826:
2825:
2799:
2795:
2789:
2785:
2779:
2775:
2770:
2767:
2766:
2747:
2745:
2742:
2741:
2725:
2722:
2721:
2695:
2691:
2685:
2681:
2675:
2671:
2666:
2663:
2662:
2645:
2641:
2639:
2636:
2635:
2618:
2614:
2612:
2609:
2608:
2607:commute. Since
2591:
2587:
2585:
2582:
2581:
2564:
2560:
2558:
2555:
2554:
2535:
2533:
2530:
2529:
2513:
2510:
2509:
2486:
2484:
2481:
2480:
2464:
2461:
2460:
2440:
2436:
2427:
2423:
2418:
2415:
2414:
2398:
2395:
2394:
2378:
2375:
2374:
2354:
2351:
2350:
2334:
2331:
2330:
2314:
2311:
2310:
2307:
2272:
2268:
2266:
2263:
2262:
2229:
2225:
2223:
2220:
2219:
2199:
2196:
2195:
2179:
2176:
2175:
2158:
2154:
2152:
2149:
2148:
2132:
2129:
2128:
2112:
2109:
2108:
2098:
2093:
2071:
2068:
2067:
2045:
2044:
2042:
2039:
2038:
2022:
2019:
2018:
1996:
1993:
1992:
1991:. Then for all
1976:
1973:
1972:
1931:
1928:
1927:
1897:
1893:
1891:
1888:
1887:
1862:
1859:
1858:
1824:
1821:
1820:
1798:
1795:
1794:
1769:
1765:
1763:
1760:
1759:
1743:
1740:
1739:
1722:
1718:
1716:
1713:
1712:
1696:
1693:
1692:
1670:
1667:
1666:
1643:
1639:
1630:
1626:
1621:
1618:
1617:
1610:
1604:
1577:
1574:
1573:
1556:
1552:
1550:
1547:
1546:
1530:
1527:
1526:
1510:
1507:
1506:
1490:
1487:
1486:
1463:
1458:
1457:
1455:
1452:
1451:
1435:
1432:
1431:
1414:
1410:
1408:
1405:
1404:
1374:
1370:
1361:
1357:
1355:
1352:
1351:
1328:
1323:
1322:
1320:
1317:
1316:
1300:
1297:
1296:
1279:
1275:
1273:
1270:
1269:
1250:
1247:
1246:
1230:
1227:
1226:
1210:
1207:
1206:
1190:
1187:
1186:
1169:
1165:
1163:
1160:
1159:
1140:
1137:
1136:
1120:
1117:
1116:
1099:
1095:
1093:
1090:
1089:
1072:
1068:
1066:
1063:
1062:
1037:
1034:
1033:
1017:
1014:
1013:
997:
994:
993:
977:
974:
973:
945:
942:
941:
904:
887:
885:
858:
854:
852:
849:
848:
829:
826:
825:
808:
804:
802:
799:
798:
776:
773:
772:
756:
753:
752:
732:
729:
728:
707:
703:
701:
698:
697:
690:
676:, a semisimple
672:, a semisimple
612:
567:
566:
565:
536:Wilhelm Killing
520:
512:
511:
510:
485:
474:
473:
472:
437:
427:
426:
425:
412:
396:
374:Dynkin diagrams
368:
358:
357:
356:
338:
316:Exponential map
305:
295:
294:
293:
274:Conformal group
253:
243:
242:
234:
226:
218:
210:
202:
183:
173:
163:
153:
134:
124:
123:
122:
103:Special unitary
47:
17:
12:
11:
5:
5997:
5987:
5986:
5981:
5976:
5962:
5961:
5952:
5933:
5913:
5912:External links
5910:
5908:
5907:
5879:
5868:Vinberg, E. B.
5864:
5859:
5846:
5833:
5818:
5791:
5774:(2): 263–276,
5763:
5750:
5735:
5730:
5715:
5702:
5684:
5656:
5653:
5646:
5643:
5639:
5634:
5628:
5624:
5618:
5614:
5610:
5607:
5602:
5597:
5593:
5577:
5557:10.1.1.128.471
5550:(3): 588–621,
5536:
5523:
5501:
5488:
5472:
5470:
5467:
5465:
5464:
5443:
5430:
5415:
5388:
5383:
5367:
5365:
5362:
5360:
5359:
5345:
5333:
5318:
5304:
5292:
5280:
5268:
5264:Humphreys 1992
5255:
5253:
5250:
5247:
5246:
5233:
5230:
5227:
5224:
5221:
5216:
5212:
5188:
5159:
5158:
5156:
5153:
5151:
5148:
5147:
5146:
5141:
5136:
5131:
5126:
5119:
5116:
5101:
5097:
5093:
5089:
5084:
5080:
5065:Dynkin diagram
5057:
5056:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
5007:
5003:
4999:
4996:
4993:
4990:
4987:
4984:
4961:
4958:
4955:
4952:
4947:
4943:
4939:
4936:
4904:
4901:
4884:
4879:
4875:
4871:
4868:
4833:Main article:
4830:
4827:
4788:Schubert cells
4773:
4772:
4761:
4758:
4755:
4750:
4747:
4744:
4740:
4736:
4733:
4711:
4690:Borel subgroup
4675:
4672:
4569:, and denoted
4529:
4525:
4521:
4518:
4513:
4509:
4505:
4502:
4486:
4460:
4459:
4445:
4442:
4439:
4436:
4433:
4429:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4375:
4348:
4296:
4293:
4280:
4260:
4257:
4254:
4251:
4247:
4243:
4240:
4237:
4234:
4214:
4210:
4206:
4203:
4200:
4197:
4177:
4174:
4171:
4168:
4164:
4160:
4157:
4154:
4151:
4131:
4111:
4108:
4105:
4102:
4080:
4076:
4055:
4052:
4049:
4029:
4009:
3985:
3982:
3979:
3976:
3973:
3953:
3942:
3941:
3929:
3926:
3923:
3920:
3916:
3912:
3909:
3906:
3903:
3900:
3897:
3874:
3871:
3868:
3865:
3862:
3859:
3848:
3847:
3835:
3831:
3827:
3824:
3821:
3818:
3815:
3812:
3789:
3769:
3749:
3738:
3737:
3725:
3722:
3719:
3716:
3707:
3704:
3699:
3696:
3692:
3688:
3685:
3681:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3630:
3627:
3624:
3621:
3601:
3581:
3566:
3565:
3553:
3550:
3547:
3544:
3535:
3532:
3529:
3524:
3521:
3517:
3513:
3510:
3506:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3455:
3452:
3449:
3446:
3426:
3406:
3382:
3358:
3338:
3326:
3323:
3298:
3295:
3261:
3237:
3226:Dynkin diagram
3212:
3204:
3196:
3187:
3180:
3162:
3159:
3146:
3126:
3106:
3103:
3083:
3063:
3043:
3031:
3030:
3018:
3015:
3010:
3006:
3000:
2996:
2990:
2986:
2982:
2961:
2958:
2937:
2926:
2914:
2911:
2906:
2902:
2896:
2892:
2886:
2882:
2878:
2857:
2854:
2833:
2822:
2810:
2807:
2802:
2798:
2792:
2788:
2782:
2778:
2774:
2753:
2750:
2729:
2718:
2706:
2703:
2698:
2694:
2688:
2684:
2678:
2674:
2670:
2648:
2644:
2621:
2617:
2594:
2590:
2567:
2563:
2541:
2538:
2517:
2492:
2489:
2468:
2448:
2443:
2439:
2435:
2430:
2426:
2422:
2402:
2382:
2371:Dynkin diagram
2358:
2338:
2318:
2309:Meanwhile, if
2306:
2303:
2290:
2287:
2284:
2280:
2275:
2271:
2250:
2247:
2244:
2241:
2237:
2232:
2228:
2216:
2215:
2203:
2183:
2161:
2157:
2136:
2116:
2097:
2096:Generating set
2094:
2092:
2089:
2088:
2087:
2075:
2052:
2049:
2026:
2006:
2003:
2000:
1980:
1962:
1961:
1935:
1915:
1912:
1909:
1905:
1900:
1896:
1872:
1869:
1866:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1808:
1805:
1802:
1772:
1768:
1747:
1725:
1721:
1711:. Recall that
1700:
1680:
1677:
1674:
1651:
1646:
1642:
1638:
1633:
1629:
1625:
1603:
1600:
1587:
1584:
1581:
1559:
1555:
1534:
1514:
1494:
1472:
1469:
1466:
1461:
1439:
1417:
1413:
1392:
1389:
1386:
1382:
1377:
1373:
1369:
1364:
1360:
1337:
1334:
1331:
1326:
1304:
1282:
1278:
1254:
1234:
1214:
1194:
1172:
1168:
1144:
1124:
1102:
1098:
1075:
1071:
1050:
1047:
1044:
1041:
1021:
1001:
981:
961:
958:
955:
952:
949:
938:
937:
925:
919:
916:
913:
910:
907:
902:
899:
896:
893:
890:
884:
881:
878:
875:
872:
869:
866:
861:
857:
833:
811:
807:
786:
783:
780:
760:
736:
710:
706:
689:
686:
644:isometry group
614:
613:
611:
610:
603:
596:
588:
585:
584:
583:
582:
577:
569:
568:
564:
563:
558:
556:Harish-Chandra
553:
548:
543:
538:
533:
531:Henri Poincaré
528:
522:
521:
518:
517:
514:
513:
509:
508:
503:
498:
493:
487:
486:
481:Lie groups in
480:
479:
476:
475:
471:
470:
465:
460:
455:
450:
445:
439:
438:
433:
432:
429:
428:
424:
423:
418:
413:
411:
410:
405:
399:
397:
395:
394:
389:
383:
381:
376:
370:
369:
364:
363:
360:
359:
355:
354:
349:
344:
339:
337:
336:
331:
325:
323:
318:
313:
307:
306:
301:
300:
297:
296:
292:
291:
286:
281:
279:Diffeomorphism
276:
271:
266:
261:
255:
254:
249:
248:
245:
244:
239:
238:
237:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
193:
192:
188:
187:
186:
185:
179:
175:
169:
165:
159:
155:
149:
142:
141:
135:
130:
129:
126:
125:
121:
120:
110:
100:
90:
80:
70:
63:Special linear
60:
53:General linear
49:
48:
43:
42:
39:
38:
30:
29:
15:
9:
6:
4:
3:
2:
5996:
5985:
5982:
5980:
5977:
5975:
5972:
5971:
5969:
5959:
5958:
5953:
5948:
5947:
5942:
5939:
5934:
5930:
5926:
5925:
5920:
5916:
5915:
5905:
5901:
5897:
5893:
5889:
5885:
5880:
5877:
5873:
5869:
5865:
5862:
5856:
5852:
5847:
5844:
5840:
5836:
5830:
5826:
5825:
5819:
5816:
5812:
5808:
5804:
5797:
5792:
5789:
5785:
5781:
5777:
5773:
5769:
5764:
5761:
5757:
5753:
5747:
5743:
5742:
5736:
5733:
5727:
5723:
5722:
5716:
5713:
5709:
5705:
5699:
5692:
5691:
5685:
5682:
5678:
5674:
5670:
5654:
5651:
5644:
5641:
5637:
5626:
5622:
5616:
5612:
5605:
5600:
5595:
5591:
5582:
5578:
5575:
5571:
5567:
5563:
5558:
5553:
5549:
5545:
5544:Ann. of Math.
5541:
5537:
5534:
5530:
5526:
5520:
5516:
5512:
5511:
5506:
5502:
5499:
5495:
5491:
5485:
5481:
5480:
5474:
5473:
5462:
5458:
5457:
5452:
5448:
5444:
5441:
5437:
5433:
5427:
5423:
5422:
5416:
5413:
5409:
5405:
5401:
5394:
5389:
5386:
5380:
5376:
5375:
5369:
5368:
5357:
5352:
5350:
5342:
5337:
5331:Theorem 11.36
5330:
5325:
5323:
5316:
5311:
5309:
5301:
5296:
5289:
5284:
5277:
5272:
5265:
5260:
5256:
5228:
5225:
5222:
5214:
5210:
5201:
5197:
5192:
5185:
5183:
5178:
5174:
5170:
5164:
5160:
5145:
5144:Hasse diagram
5142:
5140:
5137:
5135:
5134:Maximal torus
5132:
5130:
5127:
5125:
5122:
5121:
5115:
5099:
5091:
5087:
5070:
5066:
5062:
5043:
5037:
5031:
5028:
5025:
5019:
5016:
5013:
5005:
5001:
4997:
4991:
4985:
4982:
4975:
4974:
4973:
4959:
4953:
4945:
4941:
4937:
4934:
4926:
4922:
4918:
4914:
4910:
4900:
4898:
4882:
4877:
4869:
4859:
4857:
4852:
4848:
4842:
4836:
4826:
4824:
4820:
4816:
4812:
4808:
4804:
4800:
4799:Hasse diagram
4795:
4793:
4789:
4785:
4781:
4778:
4759:
4756:
4753:
4748:
4745:
4742:
4738:
4734:
4731:
4724:
4723:
4722:
4721:
4717:
4710:
4706:
4702:
4699:
4695:
4691:
4687:
4681:
4671:
4670:
4666:
4662:
4658:
4654:
4650:
4646:
4642:
4638:
4634:
4630:
4626:
4622:
4618:
4614:
4610:
4605:
4603:
4599:
4595:
4591:
4586:
4584:
4580:
4576:
4572:
4568:
4564:
4561:
4557:
4553:
4549:
4545:
4527:
4523:
4519:
4511:
4507:
4500:
4492:
4491:maximal torus
4485:
4481:
4477:
4473:
4470:is of finite
4469:
4465:
4443:
4437:
4431:
4427:
4420:
4414:
4411:
4405:
4402:
4399:
4393:
4386:
4385:
4384:
4382:
4378:
4371:
4367:
4363:
4360:of the torus
4359:
4355:
4351:
4344:
4340:
4336:
4333:of the torus
4332:
4328:
4324:
4320:
4316:
4313:
4308:
4306:
4302:
4292:
4255:
4249:
4245:
4238:
4232:
4212:
4208:
4201:
4195:
4172:
4166:
4162:
4155:
4149:
4129:
4106:
4100:
4078:
4074:
4050:
4047:
4027:
4007:
3999:
3980:
3977:
3974:
3924:
3918:
3914:
3907:
3901:
3898:
3895:
3888:
3887:
3886:
3872:
3869:
3863:
3857:
3833:
3829:
3822:
3816:
3813:
3810:
3803:
3802:
3801:
3787:
3767:
3747:
3720:
3717:
3714:
3711:for all
3705:
3702:
3697:
3694:
3690:
3686:
3683:
3675:
3672:
3669:
3663:
3657:
3651:
3644:
3643:
3642:
3625:
3619:
3599:
3579:
3571:
3548:
3545:
3542:
3539:for all
3533:
3530:
3527:
3522:
3519:
3515:
3511:
3508:
3500:
3497:
3494:
3488:
3482:
3476:
3469:
3468:
3467:
3450:
3444:
3424:
3404:
3396:
3380:
3372:
3371:maximal torus
3356:
3336:
3322:
3320:
3319:maximal torus
3316:
3312:
3308:
3304:
3294:
3292:
3288:
3287:
3282:
3278:
3273:
3259:
3252:relations in
3251:
3235:
3227:
3223:
3219:
3215:
3207:
3203:
3199:
3195:
3191:
3183:
3176:
3172:
3168:
3158:
3144:
3124:
3104:
3101:
3081:
3061:
3041:
3016:
3013:
3008:
2998:
2994:
2988:
2984:
2959:
2956:
2935:
2927:
2912:
2909:
2904:
2894:
2890:
2884:
2880:
2855:
2852:
2831:
2823:
2808:
2805:
2800:
2790:
2786:
2780:
2776:
2751:
2748:
2727:
2719:
2704:
2701:
2696:
2686:
2682:
2676:
2672:
2646:
2642:
2619:
2615:
2592:
2588:
2565:
2561:
2539:
2536:
2515:
2507:
2506:
2505:
2490:
2487:
2466:
2441:
2437:
2433:
2428:
2424:
2372:
2336:
2316:
2302:
2285:
2282:
2278:
2273:
2269:
2248:
2242:
2239:
2235:
2230:
2226:
2181:
2159:
2155:
2106:
2103:
2102:
2101:
2073:
2047:
2024:
2004:
2001:
1998:
1978:
1970:
1967:
1966:
1965:
1959:
1956:
1955:
1954:
1951:
1947:
1910:
1907:
1903:
1898:
1894:
1884:
1867:
1864:
1844:
1841:
1835:
1832:
1829:
1806:
1803:
1800:
1792:
1788:
1770:
1766:
1745:
1723:
1719:
1698:
1678:
1675:
1644:
1640:
1636:
1631:
1627:
1614:
1609:
1602:Weyl chambers
1599:
1585:
1582:
1579:
1557:
1553:
1532:
1512:
1492:
1470:
1467:
1464:
1437:
1415:
1411:
1390:
1387:
1384:
1380:
1375:
1371:
1367:
1362:
1358:
1335:
1332:
1329:
1302:
1280:
1276:
1266:
1252:
1212:
1192:
1170:
1166:
1156:
1142:
1100:
1096:
1073:
1069:
1045:
1039:
999:
979:
956:
953:
950:
923:
914:
911:
908:
897:
894:
891:
882:
879:
876:
873:
867:
859:
855:
847:
846:
845:
831:
809:
805:
781:
778:
758:
750:
708:
704:
694:
685:
683:
679:
675:
671:
666:
664:
660:
656:
652:
649:
645:
641:
637:
633:
630:(named after
629:
625:
621:
609:
604:
602:
597:
595:
590:
589:
587:
586:
581:
578:
576:
573:
572:
571:
570:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
523:
516:
515:
507:
504:
502:
499:
497:
494:
492:
489:
488:
484:
478:
477:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
440:
436:
431:
430:
422:
419:
417:
414:
409:
406:
404:
401:
400:
398:
393:
390:
388:
385:
384:
382:
380:
377:
375:
372:
371:
367:
362:
361:
353:
350:
348:
345:
343:
340:
335:
332:
330:
327:
326:
324:
322:
319:
317:
314:
312:
309:
308:
304:
299:
298:
290:
287:
285:
282:
280:
277:
275:
272:
270:
267:
265:
262:
260:
257:
256:
252:
247:
246:
235:
229:
227:
221:
219:
213:
211:
205:
203:
197:
196:
195:
194:
190:
189:
184:
182:
176:
174:
172:
166:
164:
162:
156:
154:
152:
146:
145:
144:
143:
139:
138:
133:
128:
127:
118:
114:
111:
108:
104:
101:
98:
94:
91:
88:
84:
81:
78:
74:
71:
68:
64:
61:
58:
54:
51:
50:
46:
41:
40:
36:
32:
31:
28:
24:
21:
20:
5979:Lie algebras
5956:
5944:
5922:
5887:
5883:
5875:
5871:
5850:
5823:
5806:
5802:
5771:
5767:
5740:
5720:
5689:
5675:(1): 21–25,
5672:
5668:
5547:
5543:
5509:
5478:
5454:
5451:"Weyl group"
5420:
5403:
5399:
5373:
5336:
5295:
5283:
5271:
5266:, p. 6.
5259:
5199:
5195:
5191:
5181:
5176:
5168:
5163:
5060:
5058:
4920:
4916:
4908:
4906:
4855:
4850:
4844:
4818:
4814:
4810:
4806:
4796:
4792:Grassmannian
4787:
4783:
4779:
4777:flag variety
4774:
4715:
4708:
4704:
4693:
4685:
4683:
4668:
4664:
4660:
4652:
4648:
4640:
4636:
4632:
4628:
4612:
4608:
4606:
4601:
4597:
4593:
4589:
4587:
4582:
4574:
4570:
4566:
4559:
4555:
4551:
4547:
4543:
4483:
4479:
4475:
4467:
4466:is finite –
4463:
4461:
4380:
4373:
4369:
4365:
4361:
4353:
4346:
4342:
4338:
4334:
4326:
4322:
4318:
4314:
4309:
4300:
4298:
3943:
3849:
3739:
3569:
3567:
3394:
3328:
3300:
3284:
3277:Bruhat order
3274:
3249:
3221:
3217:
3210:
3205:
3201:
3197:
3193:
3185:
3178:
3175:presentation
3164:
3032:
2308:
2217:
2127:is base for
2104:
2099:
1968:
1963:
1957:
1952:
1948:
1885:
1790:
1787:Weyl chamber
1786:
1664:
1267:
1157:
939:
726:
667:
658:
632:Hermann Weyl
627:
624:Lie algebras
617:
561:Armand Borel
546:Hermann Weyl
391:
347:Loop algebra
329:Killing form
303:Lie algebras
180:
170:
160:
150:
116:
106:
96:
86:
76:
66:
56:
27:Lie algebras
5890:: 173–186,
5809:: 155–171,
5447:Popov, V.L.
4358:centralizer
4040:. For each
3570:centralizer
3303:Lie algebra
2369:, then the
749:root system
674:Lie algebra
648:hyperplanes
636:root system
620:mathematics
541:Élie Cartan
387:Root system
191:Exceptional
5984:Lie groups
5968:Categories
5904:0136.28803
5843:0986.20038
5815:0136.28802
5788:0627.20019
5760:0483.57002
5712:1142.20020
5533:1110.05001
5498:0983.17001
5440:0725.20028
5412:1092.22004
5364:References
4911:the first
4903:Cohomology
4858:-factorial
4839:See also:
4563:Weyl group
4558:is called
4462:The group
4331:normalizer
3612:, denoted
3437:, denoted
3395:normalizer
1819:such that
1606:See also:
1598:elements.
1115:preserves
651:orthogonal
628:Weyl group
526:Sophus Lie
519:Scientists
392:Weyl group
113:Symplectic
73:Orthogonal
23:Lie groups
5946:MathWorld
5929:EMS Press
5896:2261/6049
5552:CiteSeerX
5461:EMS Press
5341:Hall 2015
5329:Hall 2015
5300:Hall 2015
5288:Hall 2015
5276:Hall 2015
5252:Citations
5184:Lie group
5150:Footnotes
5032:
5026:⋊
4998:≅
4986:
4746:∈
4739:⋃
4698:connected
4356:) by the
4312:Lie group
4279:Φ
4079:α
4054:Φ
4051:∈
4048:α
3952:Φ
3718:∈
3695:−
3673:∈
3546:∈
3528:∈
3520:−
3498:∈
3307:Lie group
3145:β
3125:α
3105:θ
3082:θ
3062:β
3042:α
2999:β
2989:α
2895:β
2885:α
2791:β
2781:α
2687:β
2677:α
2647:β
2620:α
2593:β
2566:α
2442:β
2429:α
2401:Δ
2381:Φ
2357:Δ
2337:β
2317:α
2305:Relations
2289:Φ
2286:∈
2283:α
2274:α
2246:Δ
2243:∈
2240:α
2231:α
2202:Δ
2182:α
2160:α
2135:Φ
2115:Δ
2051:¯
2002:∈
1934:Φ
1926:preserve
1914:Φ
1911:∈
1908:α
1899:α
1871:Δ
1868:∈
1865:α
1830:α
1804:∈
1771:α
1724:α
1699:α
1676:⊂
1673:Φ
1641:α
1628:α
1388:≠
1368:−
1233:Φ
1123:Φ
1101:α
1074:α
1020:Φ
957:⋅
951:⋅
924:α
915:α
909:α
898:α
880:−
860:α
832:α
810:α
785:Φ
782:∈
779:α
735:Φ
403:Real form
289:Euclidean
140:Classical
5198:acts on
5177:relative
5175:, but a
5118:See also
4835:q-analog
4701:solvable
2960:′
2856:′
2752:′
2540:′
2491:′
1857:for all
1403:, where
640:subgroup
575:Glossary
269:Poincaré
5931:, 2001
5574:1968753
4817:(where
4598:maximal
4301:defined
3998:weights
2973:, then
2869:, then
2765:, then
2553:, then
2349:are in
2105:Theorem
1969:Theorem
1958:Theorem
1525:th and
1430:is the
642:of the
638:Φ is a
634:) of a
483:physics
264:Lorentz
93:Unitary
5902:
5857:
5841:
5831:
5813:
5786:
5758:
5748:
5728:
5710:
5700:
5572:
5554:
5531:
5521:
5496:
5486:
5438:
5428:
5410:
5381:
4447:
4310:For a
3286:length
940:where
797:, let
626:, the
259:Circle
5799:(PDF)
5770:, 2,
5694:(PDF)
5671:, 1,
5570:JSTOR
5396:(PDF)
5182:split
5155:Notes
4790:(see
4786:into
4688:is a
4643:is a
4596:is a
4579:torus
4489:is a
4478:. If
4472:index
4321:<
3369:be a
2174:with
2107:: If
747:be a
334:Index
5855:ISBN
5829:ISBN
5746:ISBN
5726:ISBN
5698:ISBN
5519:ISBN
5484:ISBN
5426:ISBN
5379:ISBN
4972:as:
4663:and
4372:) =
4345:) =
3329:Let
3279:and
3250:only
3220:and
3137:and
3054:and
2948:and
2844:and
2740:and
2634:and
2580:and
2528:and
2479:and
2373:for
2329:and
1842:>
727:Let
659:most
284:Loop
25:and
5900:Zbl
5892:hdl
5839:Zbl
5811:Zbl
5784:Zbl
5776:doi
5756:Zbl
5708:Zbl
5677:doi
5667:",
5562:doi
5529:Zbl
5494:Zbl
5436:Zbl
5408:Zbl
5029:Out
4983:Out
4794:).
4692:of
4684:If
4659:of
4609:GL,
4588:If
4565:of
4560:the
4474:in
4383:),
4225:or
4093:of
3760:of
3592:in
3572:of
3417:in
3397:of
3373:in
2194:in
2066:of
1665:If
1012:of
618:In
115:Sp(
105:SU(
85:SO(
65:SL(
55:GL(
5970::
5943:.
5927:,
5921:,
5898:,
5888:11
5886:,
5876:47
5874:,
5837:,
5807:11
5805:,
5801:,
5782:,
5772:38
5754:,
5706:,
5673:10
5568:,
5560:,
5548:35
5546:,
5527:,
5513:,
5492:,
5459:,
5453:,
5434:,
5404:14
5402:,
5398:,
5348:^
5321:^
5307:^
4909:G,
4825:.
4813:-
4707:=
4669:G.
4665:Z,
4631:→
4550:=
4482:=
4412::=
4364:=
4337:=
4307:.
4291:.
3309:,
3305:,
3272:.
3213:ij
3190:=1
2301:.
1883:.
1265:.
684:.
95:U(
75:O(
5949:.
5894::
5778::
5679::
5655:1
5652:=
5645:j
5642:i
5638:k
5633:)
5627:j
5623:r
5617:i
5613:r
5609:(
5606:=
5601:2
5596:i
5592:r
5564::
5232:)
5229:T
5226:;
5223:W
5220:(
5215:1
5211:H
5200:T
5196:W
5186:.
5169:G
5100:k
5096:)
5092:2
5088:/
5083:Z
5079:(
5061:G
5044:.
5041:)
5038:G
5035:(
5023:)
5020:T
5017:;
5014:W
5011:(
5006:1
5002:H
4995:)
4992:N
4989:(
4960:,
4957:)
4954:T
4951:(
4946:G
4942:N
4938:=
4935:N
4921:T
4917:W
4883:!
4878:q
4874:]
4870:n
4867:[
4856:q
4851:n
4819:n
4815:k
4811:n
4807:k
4784:B
4782:/
4780:G
4760:B
4757:w
4754:B
4749:W
4743:w
4735:=
4732:G
4716:B
4712:0
4709:T
4705:T
4694:G
4686:B
4661:W
4653:N
4649:G
4641:N
4637:T
4635:/
4633:N
4629:N
4613:D
4602:G
4594:T
4590:G
4583:G
4575:G
4573:(
4571:W
4567:G
4556:T
4554:/
4552:N
4548:Z
4546:/
4544:N
4528:0
4524:T
4520:=
4517:)
4512:0
4508:T
4504:(
4501:Z
4487:0
4484:T
4480:T
4476:N
4468:Z
4464:W
4444:.
4441:)
4438:T
4435:(
4432:Z
4428:/
4424:)
4421:T
4418:(
4415:N
4409:)
4406:G
4403:,
4400:T
4397:(
4394:W
4381:T
4379:(
4376:G
4374:Z
4370:T
4368:(
4366:Z
4362:Z
4354:T
4352:(
4349:G
4347:N
4343:T
4341:(
4339:N
4335:N
4323:G
4319:T
4315:G
4259:)
4256:T
4253:(
4250:Z
4246:/
4242:)
4239:T
4236:(
4233:N
4213:T
4209:/
4205:)
4202:T
4199:(
4196:N
4176:)
4173:T
4170:(
4167:Z
4163:/
4159:)
4156:T
4153:(
4150:N
4130:T
4110:)
4107:T
4104:(
4101:N
4075:x
4028:K
4008:T
3984:)
3981:T
3978:,
3975:K
3972:(
3940:.
3928:)
3925:T
3922:(
3919:Z
3915:/
3911:)
3908:T
3905:(
3902:N
3899:=
3896:W
3873:T
3870:=
3867:)
3864:T
3861:(
3858:Z
3846:.
3834:T
3830:/
3826:)
3823:T
3820:(
3817:N
3814:=
3811:W
3788:T
3768:K
3748:W
3736:.
3724:}
3721:T
3715:t
3706:t
3703:=
3698:1
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3029:.
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954:,
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233:8
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119:)
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107:n
99:)
97:n
89:)
87:n
79:)
77:n
69:)
67:n
59:)
57:n
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