Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras

TL;DR

This paper explores the representation theory of three related algebra towers connected to symmetric groups and their Hecke algebras, providing detailed module structures and new interpretations of algebraic bases.

Contribution

It introduces new algebra towers, describes their module structures, and offers fresh insights into bases of quasi-symmetric and noncommutative symmetric functions.

Findings

Descriptions of simple and indecomposable projective modules

Explicit structures of Grothendieck algebras and coalgebras

New interpretations of classical algebraic bases

Abstract

We study the representation theory of three towers of algebras which are related to the symmetric groups and their Hecke algebras. The first one is constructed as the algebras generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. The two others are the monoid algebras of nondecreasing functions and nondecreasing parking functions. For these three towers, we describe the structure of simple and indecomposable projective modules, together with the Cartan map. The Grothendieck algebras and coalgebras given respectively by the induction product and the restriction coproduct are also given explicitly. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases.