The decomposition of 0-Hecke modules associated to quasisymmetric Schur functions

Sebastian König

doi:10.5802/alco.58

Abstract

Recently Tewari and van Willigenburg constructed modules of the 0-Hecke algebra that are mapped to the quasisymmetric Schur functions by the quasisymmetric characteristic and decomposed them into a direct sum of certain submodules. We show that these submodules are indecomposable by determining their endomorphism rings.

Highlights

Since the 19th century mathematicians have been interested in the Schur functions sλ and their various properties
They form an orthonormal basis of Sym, the Hopf algebra of symmetric functions and they are the images of the irreducible complex characters of the symmetric groups under the characteristic map [13]
The symmetric functions are contained in the Hopf algebra QSym of quasisymmetric functions defined by Gessel in 1984 [6]

Summary

Introduction

Since the 19th century mathematicians have been interested in the Schur functions sλ and their various properties. One such analogue due to Haglund, Luoto, Mason and van Willigenburg is given by the quasisymmetric Schur functions Sα [8] They form a basis of QSym and nicely refine the Schur functions via sλ = Sα α=λ where λ is a partition and the sum runs over all compositions α that rearrange λ [8] (see Section 2.2 for definitions). 0-Hecke algebra, composition tableau, quasisymmetric function, Schur function Another basis of QSym sharing porperties with the Schur functions is formed by the dual immaculate functions of Berg, Bergeron, Saliola, Serrano and Zabrocki [1]. Indecomposable 0-Hecke modules whose images under Ch are the dual immaculate functions were defined in [2].

Background

Findings

A 0-Hecke action on chains of the composition poset