Weyl group of a maximal torus in a linear algebraic group

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Definition

Suppose G is a linear algebraic group over a field K and T is a maximal torus in G. The Weyl group of T, denoted W(T), is defined as follows:

Note that if K is an algebraically closed field, then T is unique up to conjugacy in G, hence W(T) is uniquely determined up to isomorphism by G.

Particular cases

We denote the underlying field by K.

Case for group G Case for maximal torus T Description of Weyl group as abstract group Description of Weyl group as a subgroup of G (one of many possible)
general linear group of degree n, denoted GL(n,K) subgroup of diagonal matrices, isomorphic to n^{th} direct power of multiplicative group of K symmetric group S_n S_n embedded as permutation matrices
special linear group of degree n, denoted SL(n,K) subgroup of diagonal matrices whose determinant is one symmetric group S_n tricky, idea is that for odd permutations we make one of the entries -1.