nLab Weyl group

Not to be confused with Weil group.

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Idea

In Lie theory

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 $G$ is equivalently the quotient group of the normalizer of any maximal torus $T$ by that torus.

$$W \simeq N_G T / T \,.$$

In equivariant homotopy theory

In equivariant homotopy theory one uses the term Weyl group more generally for the quotient group

$$W_G H = (N_G H) / H$$

of the normalizer of a given subgroup $H \hookrightarrow G$ 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 $H$-fixed points of a topological G-space. (See at Change of equivariance group and fixed loci for details and, at, e.g., tom Dieck splitting for applications.)

This may be seen from the fact that the Weyl group of $H \subset G$ is the automorphism group of the coset space $G/H$ in the orbit category of $G$ (and in fact the endomorphism monoid of $G/H$, since the orbit category is an EI-category, see there):

(1)$$End_{G Orbits} \big( G/H \big) \;\; = Aut_{G Orbits} \big( G/H \big) \;\; \simeq \;\; W_G(H) \,.$$

Notice that $W_G G = 1$ and $W_G 1 = G$.

On the other hand, if $H = N \subset G$ is a normal subgroup, then its normalizer is $G$ itself, in which case the Weyl group is just the plain quotient group

$$W_G N \;\simeq\; G/N \,.$$

Definition

Given a compact Lie group $G$ with chosen maximal torus $T$, its Weyl group $W(G)=W(G,T)$ is the group of automorphisms of $T$ which are restrictions of inner automorphisms of $G$.

This is the quotient group of the normalizer subgroup of $T \subset G$ by $T$

$$W \simeq N_G(T)/T \,.$$

Properties

• The maximal torus is of finite index in its normalizer; the quotient $N(T)/T$ is isomorphic to $W(G)$.

• The cardinality of $W(G)$ for a compact connected $G$, equals the Euler characteristic of the homogeneous space $G/T$ (“flag variety”).

• An important approach to the representations of the Weyl groups is the Springer theory.

Related concepts

• Schubert calculus

References

• 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)