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

Young-Hun Kim, Dominic Searles

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.

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

Abstract

In 2001, Chow developed the theory of the posets and the type -partition enumerators . To provide a representation-theoretic interpretation of , we define the poset modules of the 0-Hecke algebra of type by endowing the set of type- linear extensions of with an -action. We then show that the Grothendieck group of the category associated to type- poset modules is isomorphic to the space of type quasisymmetric functions as both a -module and comodule, where denotes the Hopf algebra of quasisymmetric functions. Considering an equivalence relation on posets, where two posets are equivalent if they share the same set of type- 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- 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- weak Bruhat interval modules, poset modules, and finite-dimensional -modules.
Paper Structure (17 sections, 34 theorems, 188 equations, 1 algorithm)

This paper contains 17 sections, 34 theorems, 188 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter groups
  4. Quasisymmetric functions and their type $B$ analogues
  5. $0$-Hecke algebras and their representation theory
  6. Poset modules for $0$-Hecke algebras of type $A$
  7. Posets and $P$-partitions of type $B$
  8. Poset modules of type $B$ and the associated category
  9. Definition of poset modules of type $B$
  10. The category of $B_n$ poset modules and its Grothendieck group
  11. Induction of poset modules
  12. Restriction of poset modules
  13. (Anti-)automorphism twists
  14. Connection to weak Bruhat interval modules
  15. Distinguished posets
  16. ...and 2 more sections

Key Result

Theorem 2.1

(cf. DS25) Let $(W,S)$ be a Coxeter system. If $X$ is an ascent-compatible subset of $W$, then the linear operators $\overline{\uppi}_s$ define a right action of the $0$-Hecke algebra $H_W(0)$ on $\mathbb C X$.

Theorems & Definitions (78)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 3.1
  • proof
  • ...and 68 more