Equivalence classes of lower and upper descent weak Bruhat intervals

IF 1 3区 数学 Q3 MATHEMATICS, APPLIED
Seung-Il Choi , Sun-Young Nam , Young-Tak Oh
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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 Int(n) denote the set of nonempty left weak Bruhat intervals in the symmetric group Sn. We investigate the equivalence relation
on Int(n), 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 [σ,ρ]L[σ,ρ]L if and only if σRσ. 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 [w0(S),σ]L or [σ,w1(S)]L, where w0(S) is the longest element in the parabolic subgroup SS of Sn, generated by {si|iS} for a subset S[n1], and w1(S) is the longest element among the minimal-length representatives of left S[n1]S-cosets in Sn. 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 [w0(S),σ]L, an injective hull of B([w0(S),σ]L) is given, and for the equivalence class containing [σ,w1(S)]L, a projective cover of B([σ,w1(S)]L) is given. Here, B(I) denotes the weak Bruhat interval module of the 0-Hecke algebra associated with IInt(n). 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.
上下下降弱Bruhat区间的等价类
设Int(n)表示对称群Sn中非空左弱Bruhat区间的集合。研究了Int(n)上的等价关系,其中当且仅当I与j之间存在一个保持下降的偏序同构。对于的每一个等价类C,一个偏阶⪯被定义为[σ,ρ]L⪯[σ ',ρ ‘]L当且仅当σ⪯Rσ ’。Kim-Lee-Oh(2024)证明了偏序集(C,⪯)与右弱Bruhat区间同构。本文主要研究下、上下降弱Bruhat区间,特别是[w0(S),σ]L或[σ,w1(S)]L的下降弱Bruhat区间,其中w0(S)是Sn的抛物子群SS中的最长元素,由{si|i∈S}对一个子集S (n−1)产生,w1(S)是Sn中左S[n−1]∑S-集的最小长度代表中的最长元素。我们首先提供等价关系的位论表征。利用这一特征,当C是下下降区间或上下降区间时,可以确定等价类C中的最小和最大元素。在附加条件下,提供了(C,⪯)结构的详细描述。进一步,对于含有[w0(S),σ]L的等价类,给出了B([w0(S),σ]L)的一个内射壳,对于含有[σ,w1(S)]L的等价类,给出了B([σ,w1(S)]L)的一个射影覆盖。其中,B(I)表示与I∈Int(n)相关联的0-Hecke代数的弱Bruhat区间模。应用所得结果研究了0-Hecke代数的射影不可分解模的商模和子模的下下降区间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Advances in Applied Mathematics
Advances in Applied Mathematics 数学-应用数学
CiteScore
2.00
自引率
9.10%
发文量
88
审稿时长
85 days
期刊介绍: Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.
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