Classification of weak Bruhat interval modules of $0$-Hecke algebras

TL;DR

This paper classifies weak Bruhat interval modules of the $0$-Hecke algebra across all Coxeter types, providing a uniform description and resolving open questions and conjectures in the field.

Contribution

It offers a type-independent classification of these modules, answers an open question, and confirms a conjecture regarding their structure in finite Coxeter groups.

Findings

Classification of all left weak Bruhat interval modules up to isomorphism.

Equivalent, type-independent description of descent-preserving isomorphisms.

Set of minimal/maximal elements forms an interval under the right weak Bruhat order.

Abstract

Weak Bruhat interval modules of the $0$-Hecke algebra in type $A$ provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were generalized from type $A$ to all Coxeter types. In this paper, we give an equivalent description, in a type-independent manner, when two left weak Bruhat intervals in a Coxeter group are descent-preserving isomorphic. As an application, we classify all left weak Bruhat interval modules of $0$-Hecke algebras up to isomorphism, and thereby answer an open question and resolve in the affirmative a conjecture of Jung, Kim, Lee, and Oh. Additionally, for finite Coxeter groups we show that the set of minimum (or maximum) elements of all left weak Bruhat intervals in each descent-preserving isomorphism class forms an interval under the right weak Bruhat order.