{"title":"Equivalence classes of lower and upper descent weak Bruhat intervals","authors":"Seung-Il Choi , Sun-Young Nam , Young-Tak Oh","doi":"10.1016/j.aam.2025.102910","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the set of nonempty left weak Bruhat intervals in the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We investigate the equivalence relation <figure><img></figure> on <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where <figure><img></figure> if and only if there exists a descent-preserving poset isomorphism between <em>I</em> and <em>J</em>. For each equivalence class <em>C</em> of <figure><img></figure>, a partial order ⪯ is defined by <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><mi>ρ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>⪯</mo><msub><mrow><mo>[</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span> if and only if <span><math><mi>σ</mi><msub><mrow><mo>⪯</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. Kim–Lee–Oh (2024) showed that the poset <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mo>⪯</mo><mo>)</mo></math></span> is isomorphic to a right weak Bruhat interval.</div><div>In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span> or <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is the longest element in the parabolic subgroup <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, generated by <span><math><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>}</mo></math></span> for a subset <span><math><mi>S</mi><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is the longest element among the minimal-length representatives of left <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo><mo>∖</mo><mi>S</mi></mrow></msub></math></span>-cosets in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We begin by providing a poset-theoretic characterization of the equivalence relation <figure><img></figure>. Using this characterization, the minimal and maximal elements within an equivalence class <em>C</em> are identified when <em>C</em> is a lower or upper descent interval. Under an additional condition, a detailed description of the structure of <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mo>⪯</mo><mo>)</mo></math></span> is provided. Furthermore, for the equivalence class containing <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span>, an injective hull of <span><math><mi>B</mi><mo>(</mo><msub><mrow><mo>[</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>)</mo></math></span> is given, and for the equivalence class containing <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span>, a projective cover of <span><math><mi>B</mi><mo>(</mo><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>)</mo></math></span> is given. Here, <span><math><mi>B</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> denotes the weak Bruhat interval module of the 0-Hecke algebra associated with <span><math><mi>I</mi><mo>∈</mo><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. 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.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102910"},"PeriodicalIF":1.0000,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0196885825000727","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let denote the set of nonempty left weak Bruhat intervals in the symmetric group . We investigate the equivalence relation on , 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 if and only if . Kim–Lee–Oh (2024) showed that the poset 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 or , where is the longest element in the parabolic subgroup of , generated by for a subset , and is the longest element among the minimal-length representatives of left -cosets in . 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 is provided. Furthermore, for the equivalence class containing , an injective hull of is given, and for the equivalence class containing , a projective cover of is given. Here, denotes the weak Bruhat interval module of the 0-Hecke algebra associated with . 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.
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