Equivalence classes of lower and upper descent weak Bruhat intervals
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2024
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| author | Choi, Seung-Il Nam, Sun-Young Oh, Young-Tak |
| author_facet | Choi, Seung-Il Nam, Sun-Young Oh, Young-Tak |
| contents | 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 $[σ, ρ]_L \preceq [σ', ρ']_L$ if and only if $σ\preceq_R σ'$. 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 $[w_0(S), σ]_L$ or $[σ, w_1(S)]_L$, 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 $[w_0(S), σ]_L$, an injective hull of ${\sf B}([w_0(S), σ]_L)$ is given, and for the equivalence class containing $[σ, w_1(S)]_L$, a projective cover of ${\sf B}([σ, w_1(S)]_L)$ is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivalence classes of lower and upper descent weak Bruhat intervals Choi, Seung-Il Nam, Sun-Young Oh, Young-Tak Combinatorics Representation Theory 06A07, 20C08, 05E10, 05E05 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 $[σ, ρ]_L \preceq [σ', ρ']_L$ if and only if $σ\preceq_R σ'$. 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 $[w_0(S), σ]_L$ or $[σ, w_1(S)]_L$, 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 $[w_0(S), σ]_L$, an injective hull of ${\sf B}([w_0(S), σ]_L)$ is given, and for the equivalence class containing $[σ, w_1(S)]_L$, a projective cover of ${\sf B}([σ, w_1(S)]_L)$ is given. |
| title | Equivalence classes of lower and upper descent weak Bruhat intervals |
| topic | Combinatorics Representation Theory 06A07, 20C08, 05E10, 05E05 |
| url | https://arxiv.org/abs/2412.08413 |