For Lie algebra $\mathfrak{sl}_{n}$, weyl group $W$ of it is generated by $S_{\alpha_{1}},S_{\alpha_{2}},...$ where $\alpha_{i}$'s are simple roots. How can we find explicit formula for $w_{0}\in W$ by using generators such that $w_{0}(\Phi^{+})=\Phi^{-}$ and $w_{0}^{2}=I$ by using generators?
By a theorem in the book, such $w_{0}$ exists and is unique with maximal length. Since $W$ is isomorphic to symmetric group $ S_{n}$, it's equivalent to find an element in $S_{n}$ with maximal length. I don't know how to find explicit expression for such element. Is there any theorem or tool I can appeal to?
The length of an element in this setting is equal to the number of inversions of the permutation written in one line notation. So, in $S_n$, $w_0$ is the permutation defined by $w_n(k) = n+1-k$. This has length $n(n-1)/2$. The book Combinatorics of Coxeter Groups by Bjorner and Brenti is the best reference for this point of view. It's also a fun companion to other standard texts on Weyl groups that focus more on the connection to Lie theory.