Kazhdan--Lusztig cells and the Frobenius--Schur indicator

TL;DR

This paper explores the relationship between involutions in finite Coxeter groups, Kazhdan--Lusztig cells, and the Frobenius--Schur indicator, providing a new perspective on their interconnected structure.

Contribution

It demonstrates that the involution count aligns with the sum of irreducible character degrees within Kazhdan--Lusztig cells using a generalized Frobenius--Schur indicator.

Findings

Involutions correspond to sums of irreducible character degrees within cells.

A generalized Frobenius--Schur indicator for symmetric algebras is introduced.

The compatibility of involution counts with cell decomposition is established.

Abstract

Let $W$ be a finite Coxeter group. It is well-known that the number of involutions in $W$ is equal to the sum of the degrees of the irreducible characters of $W$. Following a suggestion of Lusztig, we show that this equality is compatible with the decomposition of $W$ into Kazhdan--Lusztig cells. The proof uses a generalisation of the Frobenius--Schur indicator to symmetric algebras, which may be of independent interest.