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

Yong Liao; Han Yang; Houyi Yu

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arXiv · 2410.07990

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

Yong Liao·Han Yang·Houyi Yu

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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.

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BibTeX RIS

Yong Liao, Han Yang, Houyi Yu. 2024-10-10. Classification of weak Bruhat interval modules of $0$-Hecke algebras. https://arxiv.org/abs/2410.07990

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