Abstract
In 2001, Chow developed the theory of the 𝐵𝓃 posets 𝒫 and the type 𝐵 𝒫-partition enumerators 𝒦𝐵/𝒫. To provide a representation-theoretic interpretation of 𝒦𝐵/𝒫, we define the poset modules 𝑀𝐵/𝒫 of the 0-Hecke algebra 𝐻𝐵/𝓃(0) of type 𝐵 by endowing the set of type-𝐵 linear extensions of 𝒫 with an 𝐻𝐵/𝓃(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 QSym-module and comodule, where QSym denotes the Hopf algebra of quasisymmetric functions. Considering an equivalence relation on 𝐵𝓃 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, 𝐵𝓃 poset modules, and finite-dimensional 𝐻𝐵/𝓃(0)-modules.