Weak Bruhat interval modules of the 0-Hecke algebra

TL;DR

This paper introduces weak Bruhat interval modules for the 0-Hecke algebra, unifying various tableau-based modules and analyzing their structural properties and relationships.

Contribution

It provides a unified framework for 0-Hecke modules via weak Bruhat intervals and characterizes indecomposable summands related to key quasisymmetric functions.

Findings

Indecomposable summands correspond to dual immaculate and Schur-type functions.

Weak Bruhat interval modules can be embedded into the regular representation.

The study includes induction, restriction, and involution properties.

Abstract

The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules.