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In Lie theory, a Weyl group is a group associated with a compact Lie group that can either be abstractly defined in terms of a root system or in terms of a maximal torus. More generally there are Weyl groups associated with symmetric spaces.
The Weyl group of a compact Lie group is equivalently the quotient group of the normalizer of any maximal torus by that torus.
In equivariant homotopy theory one uses the term Weyl group more general for the quotient group
of the normalizer of any subgroup by that subgroup (e.g. May 96, p. 13).
The relevance of the Weyl group in this sense is that it is the maximal group which canonically acts on -fixed points of a topological G-space. (See for instance at tom Dieck splitting.)
This may be seen from the fact that the Weyl group of is the automorphism group of the coset space in the orbit category of :
Notice that and .
On the other hand, if is a normal subgroup, then its normalizer is itself, in which case the Weyl group is just the plain quotient group
Given a compact Lie group with chosen maximal torus , its Weyl group is the group of automorphisms of which are restrictions of inner automorphisms of .
This is the quotient group of the normalizer subgroup of by
The maximal torus is of finite index in its normalizer; the quotient is isomorphic to .
The cardinality of for a compact connected , equals the Euler characteristic of the homogeneous space (“flag variety”).
An important approach to the representations of the Weyl groups is the Springer theory.
eom: Weyl group; wikipedia Weyl group
N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser 1997. x+495 pp.
Walter Borho, Robert MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707–710 MR82f:14002
Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996.
With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf, cbms-91)
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