Weyl Group - Definition

Definition

The Weyl group can be defined in various ways, depending on context (Lie algebra, Lie group, symmetric space, etc.), and a specific realization depends on a choice – of Cartan subalgebra for a Lie algebra, of maximal torus for a Lie group. The Weyl groups of a Lie group and its corresponding Lie algebra are isomorphic, and indeed a choice of maximal torus gives a choice of Cartan subalgebra.

For a Lie algebra, the Weyl group is the reflection group generated by reflections in the roots – the specific realization of the root system depending on a choice of Cartan subalgebra (maximal abelian).

For a Lie group G satisfying certain conditions, given a torus T < G (which need not be maximal), the Weyl group with respect to that torus is defined as the quotient of the normalizer of the torus N = N(T) = N_G(T) by the centralizer of the torus Z = Z(T) = Z_G(T),

The group W is finite – Z is of finite index in N. If T = T₀ is a maximal torus (so it equals its own centralizer: ) then the resulting quotient N/Z = N/T is called the Weyl group of G, and denoted W(G). Note that the specific quotient set depends on a choice of maximal torus, but the resulting groups are all isomorphic (by an inner automorphism of G), since maximal tori are conjugate. However, the isomorphism is not natural, and depends on the choice of conjugation.

For example, for the general linear group GL, a maximal torus is the subgroup D of invertible diagonal matrices, whose normalizer is the generalized permutation matrices (matrices in the form of permutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is the symmetric group. In this case the quotient map N → N/T splits (via the permutation matrices), so the normalizer N is a semidirect product of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup of G. In general this is not always the case – the quotient does not always split, the normalizer N is not always the semidirect product of N and Z, and the Weyl group cannot always be realized as a subgroup of G.