The Hecke group algebra of a Coxeter group and its representation theory

TL;DR

This paper introduces the Hecke-group algebra for finite Coxeter groups, exploring its structure, bases, and representation theory, and relates it to combinatorial objects and classical symmetric functions.

Contribution

It defines and analyzes the Hecke-group algebra, providing new descriptions, bases, and connections to combinatorics and symmetric functions, extending prior algebraic frameworks.

Findings

Detailed description of the algebra's structure and bases

Connections to monoid algebras of nondecreasing functions and parking functions

New interpretations of classical bases of quasi-symmetric and noncommutative symmetric functions

Abstract

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well. This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases.