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

Han Yang, Houyi Yu

Abstract

Weak Bruhat interval modules of the $0$-Hecke algebra in type $A$ provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were 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 left weak Bruhat interval modules of $0$-Hecke algebras up to isomorphism, and thereby answer an open question and resolve in the affirmative a conjecture of Jung, Kim, Lee, and Oh. Additionally, for finite Coxeter groups we show that the set of minimum (or maximum) elements of all left weak Bruhat intervals in each descent-preserving isomorphism class forms an interval under the right weak Bruhat order.

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

Abstract

Weak Bruhat interval modules of the -Hecke algebra in type provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were generalized from type 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 left weak Bruhat interval modules of -Hecke algebras up to isomorphism, and thereby answer an open question and resolve in the affirmative a conjecture of Jung, Kim, Lee, and Oh. Additionally, for finite Coxeter groups we show that the set of minimum (or maximum) elements of all left weak Bruhat intervals in each descent-preserving isomorphism class forms an interval under the right weak Bruhat order.

Paper Structure

This paper contains 7 sections, 28 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial orderings on Coxeter groups
  4. Weak Bruhat interval modules of $0$-Hecke algebras
  5. Colored digraph isomorphisms and descent-preserving isomorphisms
  6. Classification of weak Bruhat interval modules up to isomorphism
  7. Descent-preserving equivalence relations on ${\rm Int}(W)$

Key Result

Lemma 2.1

BB05 Let $u,v \in W$. If $u\leqslant_{B} v$ and $s \in {\rm Des}_{L}(v)\setminus {\rm Des}_{L}(u),$ then $su \leqslant_{B} v$ and $u \leqslant_{B} sv$.

Figures (3)

  • Figure 1: The weak Bruhat orders $(\mathfrak{S}_4,\leqslant_L)$ and $(\mathfrak{S}_4,\leqslant_R)$.
  • Figure 2: $H_{4}(0)$-modules $B(1234,3214)$ and $B(1243,3241)$.
  • Figure 3: The colored digraph of $[1234,3214]_L$.

Theorems & Definitions (59)

  • Conjecture 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Example 2.4
  • Example 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 49 more