{"title":"Numerical range of real-valued linear mapping on the complex Stiefel manifold: Convexity and application","authors":"Hanzhi Chen, Zhenhong Huang, Mengmeng Song, Yong Xia","doi":"10.1016/j.laa.2025.01.031","DOIUrl":"10.1016/j.laa.2025.01.031","url":null,"abstract":"<div><div>The study confirms the convexity of the joint numerical range of any <em>k</em> real-valued linear functions on the <span><math><mi>n</mi><mo>×</mo><mi>p</mi></math></span> complex Stiefel manifold under the condition <span><math><mi>k</mi><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>1</mn></math></span>. Revealing the hidden convexity of fractional linear programming on the complex Stiefel manifold, a first-time study, serves as an impactful application.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 95-110"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral extremal graphs for disjoint odd wheels","authors":"Yu Luo , Zhenyu Ni , Yanxia Dong","doi":"10.1016/j.laa.2025.01.034","DOIUrl":"10.1016/j.laa.2025.01.034","url":null,"abstract":"<div><div>For a given graph <em>F</em>, let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all <em>F</em>-free graphs of order <em>n</em>, respectively. <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> consist of the extremal graphs associated with <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, respectively. The odd wheel <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is constructed by joining a vertex to a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></math></span>. Cioabă, Desai and Tait determined the spectral extremal graphs of <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo>∉</mo><mrow><mo>{</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></mrow></math></span>. Xiao and Zamora determined the Turán number and all extremal graphs for <span><math><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, which is the union of <em>t</em> vertex-disjoint copies of <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. In this paper, we focus on the graph with maximum spectral radius among those that exclude any subgraph isomorphic to <span><math><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. We present structural characteristics of these spectral extremal graphs for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo>∉</mo><mrow><mo>{</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></mrow></math></span>. Furthermore, we demonstrate that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>∩</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>10</mn></math></span> and <em>n</em> large enough.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 243-266"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Constructing adjacency and distance cospectral graphs via regular rational orthogonal matrix","authors":"Lihuan Mao , Yuanhang Xu , Fenjin Liu , Bei Liu","doi":"10.1016/j.laa.2025.01.033","DOIUrl":"10.1016/j.laa.2025.01.033","url":null,"abstract":"<div><div>Two graphs <em>G</em> and <em>H</em> are <em>cospectral</em> if they share the same spectrum. Constructing <em>cospectral</em> non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature. In this paper, we construct infinite families of adjacency cospectral graphs through the GM-switching method based on generalized Johnson graphs. We give some graph operations (e.g. rooted-product, corona, cartesian product, and coalescence) to construct distance cospectral graphs with different edges via a regular rational orthogonal matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 111-128"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polynomial codimension growth of superalgebras with superautomorphism","authors":"Sara Accomando","doi":"10.1016/j.laa.2025.01.035","DOIUrl":"10.1016/j.laa.2025.01.035","url":null,"abstract":"<div><div>In this paper we present some results concerning associative superalgebras endowed with a superautomorphism of order ≤2. We characterize the superalgebras with superautomorphism with multiplicities of the cocharacter bounded by a constant. Moreover, we determine the characterization of the superalgebras with superautomorphism with polynomial growth of the codimensions and we give a classification of the subvarieties of the varieties of almost polynomial growth. Finally, we characterize the superalgebras with superautomorphism with linear growth of the codimensions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 50-79"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143137954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S.P. Glasby , Alice C. Niemeyer , Cheryl E. Praeger
{"title":"Bipartite q-Kneser graphs and two-generated irreducible linear groups","authors":"S.P. Glasby , Alice C. Niemeyer , Cheryl E. Praeger","doi":"10.1016/j.laa.2025.01.032","DOIUrl":"10.1016/j.laa.2025.01.032","url":null,"abstract":"<div><div>Let <span><math><mi>V</mi><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> be a <em>d</em>-dimensional vector space over the field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of order <em>q</em>. Fix positive integers <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> satisfying <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mi>d</mi></math></span>. Motivated by analysing a fundamental algorithm in computational group theory for recognising classical groups, we consider a certain quantity <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> which arises in both graph theory and group representation theory: <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> is the proportion of 3-walks in the ‘bipartite <em>q</em>-Kneser graph’ <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> that are closed 3-arcs. We prove that, for a group <em>G</em> satisfying <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>⊴</mo><mi>G</mi><mo>⩽</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, the proportion of certain element-pairs in <em>G</em> called ‘<span><math><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-stingray duos’ which generate an irreducible subgroup is also equal to <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. We give an exact formula for <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, and prove that<span><span><span><math><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo><</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo><</mo><mn>1</mn","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 203-229"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Rao-Mitra-Bhimasankaram relation is strongly antisymmetric","authors":"Oskar Maria Baksalary , Dennis Bernstein","doi":"10.1016/j.laa.2025.01.029","DOIUrl":"10.1016/j.laa.2025.01.029","url":null,"abstract":"<div><div>For <span><math><mi>n</mi><mo>×</mo><mi>m</mi></math></span> complex matrices <em>A</em> and <em>B</em>, the Rao-Mitra-Bhimasankaram (RMB) relation <span><math><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover></math></span>, defined by <span><math><mi>A</mi><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover><mi>B</mi></math></span> if <span><math><mi>A</mi><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mo>=</mo><mi>A</mi><msup><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi></math></span>, is reflexive and antisymmetric, but not transitive. This paper shows that, despite the lack of transitivity, <span><math><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover></math></span> is strongly antisymmetric in the sense that, for all integers <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover><mo>⋯</mo><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> implies <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The proof of this result is based on a novel proof that <span><math><mover><mrow><mo>≤</mo></mrow><mrow><mi>RMB</mi></mrow></mover></math></span> is antisymmetric.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 80-94"},"PeriodicalIF":1.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rosenbrock's theorem on system matrices over elementary divisor domains","authors":"Froilán M. Dopico , Vanni Noferini , Ion Zaballa","doi":"10.1016/j.laa.2025.01.028","DOIUrl":"10.1016/j.laa.2025.01.028","url":null,"abstract":"<div><div>Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix <em>G</em> with the Smith form of an irreducible polynomial system matrix <em>P</em> giving rise to <em>G</em> and the Smith form of a submatrix of <em>P</em>. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices <em>P</em> with entries in an arbitrary elementary divisor domain <span><math><mi>R</mi></math></span> and matrices <em>G</em> with entries in the field of fractions of <span><math><mi>R</mi></math></span>. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix <em>P</em> itself has entries in the field of fractions of the elementary divisor domain.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 10-49"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"When does subtracting a rank-one approximation decrease tensor rank?","authors":"Emil Horobeţ , Ettore Teixeira Turatti","doi":"10.1016/j.laa.2025.01.025","DOIUrl":"10.1016/j.laa.2025.01.025","url":null,"abstract":"<div><div>Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 397-415"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the nullity of middle graphs","authors":"Xinmei Yuan , Danyi Li , Weigen Yan","doi":"10.1016/j.laa.2025.01.030","DOIUrl":"10.1016/j.laa.2025.01.030","url":null,"abstract":"<div><div>Let <em>G</em> be a connected graph, and let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the line graph and middle graph of <em>G</em>. Gutman and Sciriha (On the nullity of line graphs of trees, Discrete Mathematics, 232 (2001), 35-45) proved that the nullity <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo></math></span> of <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> of a tree <em>T</em> satisfies <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. But the problem to determine which trees <em>T</em> satisfy <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> is still open. In this paper, we prove that <span><math><mi>η</mi><mo>(</mo><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> if <em>G</em> is a bipartite graph, and <span><math><mi>η</mi><mo>(</mo><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> otherwise. As an application, we show that <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> for the so-called silicate network <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> obtained from the hexagonal lattice in the context of statistical physics.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 1-9"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The dimension formula for certain twisted Jacquet modules of a cuspidal representation of GL(n,Fq)","authors":"Kumar Balasubramanian , Himanshi Khurana","doi":"10.1016/j.laa.2025.01.027","DOIUrl":"10.1016/j.laa.2025.01.027","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a positive integer. Let <em>F</em> be the finite field of order <em>q</em> and <span><math><mi>G</mi><mo>=</mo><mi>GL</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. Let <span><math><mi>P</mi><mo>=</mo><mi>M</mi><mi>N</mi></math></span> be the standard parabolic subgroup of <em>G</em> corresponding to the partition <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo></math></span>. Let <span><math><mi>A</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>×</mo><mi>k</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be a rank <em>t</em> matrix. In this paper, we compute the dimension formula for the twisted Jacquet module <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow></msub></math></span> that depends on <span><math><mi>n</mi><mo>,</mo><mi>k</mi></math></span> and <em>t</em>, when <em>π</em> is an irreducible cuspidal representation of <em>G</em> and <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is a character of <em>N</em> associated with <em>A</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 151-164"},"PeriodicalIF":1.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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