Is there a quick way to find the longest element of a subgroup of a Weyl group?

Question

Suppose $W$ is a Weyl group generated by a set of reflections $S$. For $I\subset S$, there are subgroups $W_I$ generated by the reflections $s\in I$. If $w_I$ denotes the longest element of $W_I$, is there a known formula to write down $w_I$ in terms of the $s\in I$?

Answer 1

The subgroup $W_I$ is a so-called standard parabolic subgroup. It is a Weyl group with set of generators $I$, and the length function in $W_I$ is the restriction of the length function on $W$. The group $W_I$ may not be irreducible. So you want a formula for the longest element in a group which is a product of Weyl groups and the question may be formulated independently of the bigger group $W$. The longest element will be given by the product of the longest elements in each factors and the lengths of those add. Hence what you have to do is to find the longest element in each factor. To this end one first has to identify the decomposition of $W_I$ as a direct product of irreducible subgroups and then use formulas for the various longest elements of the factors depending on their types.