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
junhu303@qq.com
Jing Zhang
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China
ellenbox@bit.edu.cn
[Abstract-pdf]
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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