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 $\stackrel{D}{\simeq }$ on $\mathrm{Int}(n)$, where $I\stackrel{D}{\simeq }J$ if and only if there exists a descent-preserving poset isomorphism between $I$ and $J$. For each equivalence class $C$ of $(\mathrm{Int}(n),\stackrel{D}{\simeq })$, a partial order $⪯$ is defined by $[\ddot{I}ƒ,\ddot{I}{]}_L⪯[\ddot{I}{ƒ}^{\prime },\ddot{I}{}^{\prime }{]}_L$ if and only if $\ddot{I}ƒ{⪯}_R\ddot{I}{ƒ}^{\prime }$. Kim-Lee-Oh (2023) 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 $[w_0(S),\ddot{I}ƒ{]}_L$ or $[\ddot{I}ƒ,w_1(S){]}_L$, 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 $\stackrel{D}{\simeq }$. 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 $(C,⪯)$ is provided. Furthermore, for the equivalence class containing $[w_0(S),\ddot{I}ƒ{]}_L$, an injective hull of $\mathsf{B}([w_0(S),\ddot{I}ƒ{]}_L)$ is given, and for the equivalence class containing $[\ddot{I}ƒ,w_1(S){]}_L$, a projective cover of $\mathsf{B}([\ddot{I}ƒ,w_1(S){]}_L)$ is given.

49 pages