Abstract:

Let $(W,S)$ be a Coxeter system and $\theta: W \to W$ an involutive automorphism which preserves $S$. A {\em twisted identity} is an element of the form $\theta (w^{-1})w$ for $w \in W$.

One aim of this talk is to convince the audience that it is well-motivated to study the set of twisted identities, and in particular the poset which these elements induce under the Bruhat order on $W$. One reason is a close connection between this poset and certain "Bruhat decompositions" of symmetric varieties. As an example, a natural cell decomposition of $SL_{2n}/Sp_{2n}$ is governed by the Bruhat order on the twisted identities in the symmetric group $S_{2n}$ when $\theta$ is "conjugation by the reverse permutation".

We present results (partly conjectural) on combinatorial and topological properties of the Bruhat order on twisted identities as well as some intriguing identities involving the Poincar\'e series.

This is work in progress.