The Weyl Group of GL(n, C)

Here’s a cleaner explanation of the Weyl group of GL(n) than I’ve seen before.  I came up with this myself, but it’s straightforward enough that I’m sure I’m not the first.

Let V be an n-dimensional complex vector space, and fix a basis \beta := \{ e_1, e_2, \ldots, e_n \} for V.  Write G := GL(V) \cong GL_n(\mathbb{C}).  Let T < G denote the subgroup of matrices which are diagonal in the basis \beta; this is a maximal torus.  We know that the Weyl group is isomorphic to N(T)/T, so let’s determine N(T).

Pick a matrix D \in T all of whose eigenvalues are distinct, and suppose A \in N(T).  Then A^{-1} D A is diagonal.  This means that A represents a change of basis from \beta to some basis in which D is diagonal.  Now D is diagonal in some basis iff that basis consists of eigenvectors of D.  Since D was chosen in such a way that its eigenspaces are one-dimensional, the only eigenvectors of D are nonzero scalar multiples of the e_i.  Therefore we have A = P C, where P is a permutation matrix and C is diagonal.  Conversely, it’s easy to see that any such matrix normalizes T.  

From here it’s clear that N(T)/T \cong S_n, since the cosets correspond to permutation matrices.

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