Maximal Coset representative for the Weyl group of a Parabolic

Question

Let $G=SL_n$ and let $P_i$ be a maximal parabolic corresponding to a simple root say $\alpha_i$. Let $W_{P_i}$ be the Weyl group of $P_i$. Is there an efficient way to compute the longest coset representative $w^{max}$ for $wP_i$ ? How can we go from $w^{min}$ to $w^{max}$ ? It seems there is a command to find the minimal representative in Macaulay-2 but not for the maximal one.

Answer 1

The Weyl group is the symmetric group $S_n$. The Weyl group of $P_i$ is $S_i\times S_{n-i}$ acting on the right. Let $$ w=(a_1,\ldots,a_i,a_{i+1},\ldots,a_n)\in S_n $$ Then $w^{min}$ (or $w^{max}$) is obtained by bringing the first $i$ and the last $n-i$ entries into an increasing (or decreasing, respectively) order. So if $i=2$ and $w=(3,1,2,4)$ then $w^{min}=(1,3,2,4)$ and $w^{max}=(3,1,4,2)$.