A subalgebra of 0-Hecke algebra

TL;DR

This paper generalizes a monoid structure on subsets of Coxeter groups, originally constructed for Weyl groups, to all finite Coxeter groups using elementary combinatorics, and provides explicit calculations for each type.

Contribution

It extends the monoid construction from Weyl groups to all finite Coxeter groups with an elementary combinatorial approach and explicit type-by-type calculations.

Findings

Monoid structure is generalized to all finite Coxeter groups.

Explicit calculations of the monoid for each Coxeter group type.

Elementary combinatorial methods are used for the generalization.

Abstract

Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $\chi$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.