The $q$-Poincaré sum and separability in a Coxeter group

Abstract

Given a finite Coxeter group $W$ and an element $q$ of a commutative ring $\mathbf{k}$, we define the element ${\mathbf{L}}_q:={\sum }_{w\in W}{q}^{\ell \left(w\right)}w$ of its group algebra $\mathbf{k}\left[W\right]$. We say that an element $w\in W$ is $W$\emph{-length-symmetric} if its coefficient in the commutator $\left[{\mathbf{L}}_a,{\mathbf{L}}_b\right]$ is $0$ for all $\mathbf{k}$ and all $a,b\in \mathbf{k}$. We prove several sufficient criteria for elements to be $W$-length-symmetric. In particular, we show that any separable permutation (i.e., permutation avoiding the patterns $2413$ and $3142$) in the symmetric group $S_n$ is $S_n$-length-symmetric. More generally, we construct a notion of separable elements of a finite Coxeter group $W$ (which, for finite Weyl groups, agrees with the eponymous notion introduced by Gaetz and Gao), and show that all such elements are $W$-length-symmetric. This is only a sufficient condition; other $W$-length-symmetric elements include (twisted) involutions and elements of dihedral parabolic subgroups.

arXiv comment39 pages. Comments are welcome!