Equivalence classes of lower and upper descent weak Bruhat intervals

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.