The biHecke monoid of a finite Coxeter group

TL;DR

This paper introduces the biHecke monoid for finite Coxeter groups, analyzing its structure and representation theory, and providing combinatorial models for its simple modules, extending previous algebraic frameworks.

Contribution

It constructs and studies the biHecke monoid generated by bubble sort and antisort operators for finite Coxeter groups, including a combinatorial model for simple modules.

Findings

The biHecke monoid has |W| simple and projective modules.

A combinatorial module for each w in W supports the simple modules.

The representation theory generalizes that of the Hecke group algebra.

Abstract

The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it has |W| simple and projective modules. In order to construct a combinatorial model for the simple modules, we introduce for each w in W a combinatorial module whose support is the interval [1,w] in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra. This involves the introduction of a w-analogue of the combinatorics of descents of W and a generalization to finite Coxeter groups of blocks of permutation matrices.