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 for V. Write . Let T < G denote the subgroup of matrices which are diagonal in the basis ; 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 all of whose eigenvalues are distinct, and suppose . Then is diagonal. This means that A represents a change of basis from 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 . 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 , since the cosets correspond to permutation matrices.