Equivalence classes of lower and upper descent weak Bruhat intervals

Abstract

Let $\mathrm{Int}\left(n\right)$ denote the set of nonempty left weak Bruhat intervals in the symmetric group $S_n$. We investigate the equivalence relation

on $\mathrm{Int}\left(n\right)$, where

if and only if there exists a descent-preserving poset isomorphism between I and J. For each equivalence class C of

, a partial order ⪯ is defined by ${[\sigma ,\rho ]}_L⪯{[{\sigma }^{\prime },{\rho }^{\prime }]}_L$ if and only if $\sigma {⪯}_R{\sigma }^{\prime }$. Kim–Lee–Oh (2024) showed that the poset $(C,⪯)$ 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 ${\left[w_0\right(S),\sigma ]}_L$ or ${[\sigma ,w_1(S\left)\right]}_L$, where $w_0\left(S\right)$ is the longest element in the parabolic subgroup $S_S$ of $S_n$, generated by $\left\{s_i\right|i\in S\}$ for a subset $S\subseteq [n-1]$, and $w_1\left(S\right)$ is the longest element among the minimal-length representatives of left $S_{[n-1]\setminus S}$-cosets in $S_n$. We begin by providing a poset-theoretic characterization of the equivalence relation

. 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 $(C,⪯)$ is provided. Furthermore, for the equivalence class containing ${\left[w_0\right(S),\sigma ]}_L$, an injective hull of $B\left({\left[w_0\right(S),\sigma ]}_L\right)$ is given, and for the equivalence class containing ${[\sigma ,w_1(S\left)\right]}_L$, a projective cover of $B\left({[\sigma ,w_1(S\left)\right]}_L\right)$ is given. Here, $B\left(I\right)$ denotes the weak Bruhat interval module of the 0-Hecke algebra associated with $I\in \mathrm{Int}\left(n\right)$. The results obtained are applied to investigate lower descent intervals arising from quotient modules and submodules of projective indecomposable modules of the 0-Hecke algebra.