The biHecke monoid of a finite Coxeter group and its representations

TL;DR

This paper introduces the biHecke monoid and cutting poset for finite Coxeter groups, exploring their structure and representation theory, including simple modules and combinatorial models, extending previous work on Hecke algebras.

Contribution

It constructs the biHecke monoid and cutting poset for finite Coxeter groups, analyzing their representation theory and introducing new combinatorial modules.

Findings

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

The module T_w is supported on the interval [1,w]_R in right weak order.

Representation theory generalizes that of the Hecke group algebra with new combinatorial structures.

Abstract

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. 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 admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w in W a combinatorial module T_w whose support is the interval [1,w]_R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.