Twisted identities in Coxeter groups.

Hultman, Axel

Journal of Algebraic Combinatorics, Tome 28 (2008) no. 2, pp. 313-332.

Résumé

Summary: Given a Coxeter system $( W, S)$ equipped with an involutive automorphism $\theta $, the set of twisted identities is $i( q)={ q( w ^{ -1}) w | w$ Ĩ $W}. \iota (\theta )=\{\theta(w^{-1})$w$\mid w\in $W}. We point out how $\iota ( \theta )$ shows up in several contexts and prove that if there is no $s\in S$ such that $s \theta ( s)$ is of odd order greater than 1, then the Bruhat order on $\iota ( \theta )$ is a graded poset with rank function $\rho $ given by halving the Coxeter length. Under the same condition, it is shown that the order complexes of the open intervals either are PL spheres or $\Bbb $Z-acyclic. In the general case, contractibility is shown for certain classes of intervals. Furthermore, we demonstrate that sometimes these posets are not graded. For the Poincaré series of $\iota ( \theta )$, i.e. its generating function with respect to $\rho $, a factorisation phenomenon is discussed.

Détail

Keywords: keywords Coxeter groups, Bruhat order, twisted identities, twisted involutions