A tableau approach to the representation theory of 0-Hecke algebras

TL;DR

This paper introduces a tableau-based method for studying the representation theory of 0-Hecke algebras across types A, B, and D, linking algebraic structures to symmetric functions.

Contribution

It extends tableau approaches to 0-Hecke algebras of types B and D, connecting their representations to quasisymmetric and noncommutative symmetric functions.

Findings

Established a tableau approach for type A 0-Hecke algebras.

Extended the approach to types B and D.

Linked representations to symmetric functions of types B and D.

Abstract

A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob--Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to type B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of type B and D and quasisymmetric functions and noncommutative symmetric functions of type B and D. Other applications are also provided.