Action of Weyl group on $GL_n$

Question

Let $K$ be a field with $\operatorname{char}(K)=0$ and $G=\operatorname{GL}_n$ defined over $K$. The Weyl group $W$ of $G$ is isomorphic to the symmetric group $S_n$. I'm now wondering if $W$ acts naturally from left on $G$ by permuting rows or columns?

Answer 1

There are many possible embeddings of the Weyl group into $G$, because it is just a complement to $T$ in $N_G(T)$. A 'canonical'${}^1$ embedding is to send $\sigma\in S_n$ to the permutation matrix corresponding to $\sigma$. Whether it permutes rows or columns therefore depends on whether your action of $\mathrm{GL}_n(K)$ on the vector space is a left or a right action.

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${}^1$ It's a good question as to whether it's possible to have a canonical anything, by definition of 'canonical'. Perhaps it's better to say 'natural' here.