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.