Equivalence classes of lower and upper descent weak Bruhat intervals

TL;DR

This paper studies the structure of equivalence classes of certain weak Bruhat intervals in the symmetric group, characterizing their poset structure and providing algebraic covers for specific classes.

Contribution

It offers a poset-theoretic characterization of descent-based equivalence classes and describes their minimal, maximal, and algebraic cover structures for lower and upper descent intervals.

Findings

Characterization of equivalence classes via poset theory

Identification of minimal and maximal elements in classes

Construction of injective and projective covers for specific classes

Abstract

Let $\mathrm{Int}(n)$ denote the set of nonempty left weak Bruhat intervals in the symmetric group $\mathfrak{S}_n$. We investigate the equivalence relation $\overset{D}{\simeq}$ on $\mathrm{Int}(n)$, where $I \overset{D}{\simeq} J$ if and only if there exists a descent-preserving poset isomorphism between $I$ and $J$. For each equivalence class $C$ of $(\mathrm{Int}(n), \overset{D}{\simeq})$, a partial order $\preceq$ is defined by $[\sigma, \rho]_L \preceq [\sigma', \rho']_L$ if and only if $\sigma \preceq_R \sigma'$. Kim-Lee-Oh (2023) showed that the poset $(C, \preceq)$ is isomorphic to a right weak Bruhat interval. In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form $[w_0(S), \sigma]_L$ or $[\sigma, w_1(S)]_L$, where $w_0(S)$ is the longest element in the parabolic subgroup $\mathfrak{S}_S$ of $\mathfrak{S}_n$, generated by $\{s_i \mid i \in S\}$ for a subset $S \subseteq [n-1]$, and $w_1(S)$ is the longest element among the minimal-length representatives of left $\mathfrak{S}_{[n-1] \setminus S}$-cosets in $\mathfrak{S}_n$. We begin by providing a poset-theoretic characterization of the equivalence relation $\overset{D}{\simeq}$. Using this characterization, the minimal and maximal elements within an equivalence class $C$ are identified when $C$ is a lower or upper descent interval. Under an additional condition, a detailed description of the structure of $(C, \preceq)$ is provided. Furthermore, for the equivalence class containing $[w_0(S), \sigma]_L$, an injective hull of ${\sf B}([w_0(S), \sigma]_L)$ is given, and for the equivalence class containing $[\sigma, w_1(S)]_L$, a projective cover of ${\sf B}([\sigma, w_1(S)]_L)$ is given.