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Journal of Lie Theory 27 (2017), No. 3, 671--706
Copyright Heldermann Verlag 2017

On Involutions in Weyl Groups

Jun Hu
Dept. of Mathematics, Beijing Institute of Technology, Beijing 100081, P. R. China

Jing Zhang
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China


Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set of twisted involutions relative to $\ast$ in $W$. In this paper we consider the case when $\ast={\rm id}$ and study the braid $I_\ast$-transformations between the reduced $I_\ast$-expressions of involutions. If $W$ is the Weyl group of type $B_n$ or $D_n$, we explicitly describe a finite set of basic braid $I_\ast$-transformations for all $n$ simultaneously, and show that any two reduced $I_\ast$-expressions for a given involution can be transformed into each other through a series of basic braid $I_\ast$-transformations. In both cases, these basic braid $I_\ast$-transformations consist of the usual basic braid transformations plus some natural ``right end transformations" and exactly one extra transformation. The main result generalizes our previous work for the Weyl group of type $A_{n}$.

Keywords: Weyl groups, Hecke algebras, twisted involutions.

MSC: 20F55; 20C08

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