The Weyl group of the Lie algebra $\mathfrak{sl}_n$ is just the symmetric group on $n$ elements, $S_n$. The action can be realized as follows. If $\mathfrak{h}$ is the Cartan subalgebra of all diagonal matrices with trace zero, then $S_n$ acts on $\mathfrak{h}$ via conjugation by permutation matrices. This action induces an action on the dual space $\mathfrak{h}^\ast$, which is the required Weyl group action. Weyl group – Wiki
question 1: "$S_n$ acts on $\mathfrak{h}$ via conjugation by permutation matrices", what would that action?
question 2: "This action induces an action on the dual space", how would you describe this induced action?
Thank you very much
Best Answer
For question 2, whenever a group $G$ acts on a vector space $V$, it also acts on the dual space $V^*$ as follows. If $g\in G$ and $\alpha\in V^*$ then $g\alpha$ is the element of $V^*$ that (regarded as a function from $V$ to scalars) sends each $v\in V$ to $\alpha(g^{-1}v)$.
Perhaps a nicer way to remember the formula is the equivalent form $(g\alpha)(gv)=\alpha(v)$.