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

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.

💡 Research Summary

This paper develops a comprehensive representation‑theoretic framework linking type B posets, type B P‑partition enumerators, and the 0‑Hecke algebra of type B. Building on Chow’s theory of Bₙ‑posets and the associated type B P‑partition generating functions Kᴮ_P, the authors introduce for each Bₙ‑poset P a right Hₙᴮ(0)‑module Mᴮ_P. The underlying vector space of Mᴮ_P is spanned by the type‑B linear extensions of P, i.e., signed permutations that respect the poset relations and are themselves Bₙ‑posets. The action of the 0‑Hecke generators π_s is defined by right multiplication in the right weak Bruhat order: π_s sends a linear extension σ to σ·s if σ·s remains a linear extension, and to zero otherwise. This action satisfies the defining relations of Hₙᴮ(0), making Mᴮ_P a well‑defined module.

The central result is that the type‑B quasisymmetric characteristic chᴮ(Mᴮ_P) coincides exactly with the type‑B P‑partition enumerator Kᴮ_P. The proof proceeds by expanding Kᴮ_P as a sum over ℒᴮ(P) and showing that the π_s‑action on the basis elements mirrors the variable substitutions inherent in the definition of Kᴮ_P. Consequently, the Grothendieck group G₀(Pᴮ) of the category of all such poset modules is isomorphic to the algebra QSymᴮ of type‑B quasisymmetric functions. Moreover, this isomorphism respects both the QSym‑module structure (induction product) and the QSym‑comodule structure (coaction), establishing a full Hopf‑module correspondence.

Induction and restriction formulas are derived for these modules. For posets P₁∈B_m and P₂∈B_n, the induced module Ind_{H_mᴮ⊗H_nᴮ}^{H_{m+n}ᴮ}(Mᴮ_{P₁}⊗Mᴮ_{P₂}) is shown to be isomorphic to Mᴮ_{P₁⊔P₂}, where ⊔ denotes the disjoint union of posets. Restriction to a smaller 0‑Hecke algebra decomposes a module according to the intersection of its linear extensions with the appropriate sub‑signed‑permutation groups. These operations translate directly into the QSym‑module product and coproduct, confirming that the Grothendieck ring of poset modules reproduces the Hopf algebra structure of QSymᴮ.

The paper also investigates (anti‑)automorphism twists of the module category. The involution θ fixing each π_s leaves Mᴮ_P unchanged up to isomorphism, while the anti‑automorphism φ sending π_s to –π_s transforms Mᴮ_P into the module associated with the opposite poset P^{op}. These results parallel known type‑A phenomena and demonstrate the robustness of the construction under categorical symmetries.

A key combinatorial contribution is the introduction of an equivalence relation on Bₙ‑posets: two posets are equivalent if they share the same set of type‑B linear extensions. Within each equivalence class the authors select a canonical representative, termed a distinguished poset. They prove that every poset module is isomorphic to the module of its distinguished representative, thereby reducing the study to a well‑behaved subclass.

Regular posets are defined as those whose linear extensions form an interval in the right weak Bruhat order on the hyperoctahedral group Sᴮₙ. Extending Bjorner–Wachs’s type‑A regular poset theory, the authors show that a distinguished poset is regular precisely when its linear extensions constitute such an interval, and conversely every interval arises from a unique regular poset. Consequently, regular poset modules coincide with type‑B weak Bruhat interval modules, providing a type‑B analogue of the interval‑module theory developed for type A.

Finally, the Grothendieck group of regular poset modules (equivalently, of type‑B weak Bruhat interval modules) is proved to be isomorphic to QSymᴮ as both a QSym‑module and a QSym‑comodule. This establishes a complete type‑B counterpart to the type‑A results of earlier work, linking combinatorial interval structures, representation theory of 0‑Hecke algebras, and the algebraic theory of quasisymmetric functions.

Overall, the paper delivers a unified picture: type‑B poset modules furnish a concrete representation‑theoretic model for type‑B quasisymmetric functions, their induction/restriction behavior mirrors the Hopf algebra operations, and the classification via distinguished and regular posets connects these modules to Bruhat interval combinatorics. The work opens avenues for further exploration of type‑B analogues of known type‑A phenomena, such as categorifications, connections to other Hecke specializations, and the study of new bases in QSymᴮ.