Poset modules of the $0$-Hecke algebras of type $B$

TL;DR

This paper introduces poset modules for the 0-Hecke algebra of type B, linking combinatorial poset structures with algebraic representations and quasisymmetric functions, and characterizing distinguished posets within this framework.

Contribution

It defines type B poset modules for the 0-Hecke algebra, establishes their connection to quasisymmetric functions, and characterizes distinguished posets with interval linear extensions.

Findings

Grothendieck group is isomorphic to type B quasisymmetric functions

Identifies distinguished posets with linear extensions forming Bruhat order intervals

Explores relationships among various module categories

Abstract

In 2001, Chow developed the theory of the $B_n$ posets $P$ and the type $B$ $P$-partition enumerators $K^B_P$. To provide a representation-theoretic interpretation of $K^B_P$, we define the poset modules $M^B_P$ of the 0-Hecke algebra $H_n^B(0)$ of type $B$ by endowing the set of type-$B$ linear extensions of $P$ with an $H_n^B(0)$-action. We then show that the Grothendieck group of the category associated to type-$B$ poset modules is isomorphic to the space of type $B$ quasisymmetric functions as both a $\mathrm{QSym}$-module and comodule, where $\mathrm{QSym}$ denotes the Hopf algebra of quasisymmetric functions. Considering an equivalence relation on $B_n$ posets, where two posets are equivalent if they share the same set of type-$B$ linear extensions, we identify a natural representative of each equivalence class, which we call a distinguished poset. We further characterize the distinguished posets whose sets of type-$B$ linear extensions form intervals in the right weak Bruhat order on the the hyperoctahedral groups. Finally, we discuss the relationship among the categories associated to type-$B$ weak Bruhat interval modules, $B_n$ poset modules, and finite-dimensional $H_n^B(0)$-modules.