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

Abstract

Weak Bruhat interval modules of the $0$-Hecke algebra in type $A$ offer a unified framework for studying modules associated to quasisymmetric functions. This class of modules has recently been 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 weak Bruhat interval modules of $0$-Hecke algebras up to isomorphism, and thereby answering an open question of Jung-Kim-Lee-Oh and confirming a conjecture of Kim-Lee-Oh. Furthermore, for finite Coxeter groups, we show that the set of minimum (respectively, maximum) elements of all left weak Bruhat intervals within each descent-preserving isomorphism class forms an interval under the right weak Bruhat order.