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Hoffman colorings of graphs
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.036
Aida Abiad , Wieb Bosma , Thijs van Veluw
Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for general graphs not much is known. We investigate tightness of the Hoffman bound, with a particular focus on irregular graphs, obtaining several results on the graph structure of Hoffman colorings. In particular, we prove a Decomposition Theorem, which characterizes the structure of Hoffman colorings, and we use it to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, leading to an algorithm for computing all connected Hoffman colorable graphs for some given number of vertices and colors. Since several graph coloring parameters are known to be sandwiched between the Hoffman bound and the chromatic number, as a byproduct of our results, we obtain the values of these chromatic parameters.
{"title":"Hoffman colorings of graphs","authors":"Aida Abiad ,&nbsp;Wieb Bosma ,&nbsp;Thijs van Veluw","doi":"10.1016/j.laa.2025.01.036","DOIUrl":"10.1016/j.laa.2025.01.036","url":null,"abstract":"<div><div>Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for general graphs not much is known. We investigate tightness of the Hoffman bound, with a particular focus on irregular graphs, obtaining several results on the graph structure of Hoffman colorings. In particular, we prove a Decomposition Theorem, which characterizes the structure of Hoffman colorings, and we use it to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, leading to an algorithm for computing all connected Hoffman colorable graphs for some given number of vertices and colors. Since several graph coloring parameters are known to be sandwiched between the Hoffman bound and the chromatic number, as a byproduct of our results, we obtain the values of these chromatic parameters.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 129-150"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138085","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}
引用次数: 0
Numerical range of real-valued linear mapping on the complex Stiefel manifold: Convexity and application
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.031
Hanzhi Chen, Zhenhong Huang, Mengmeng Song, Yong Xia
The study confirms the convexity of the joint numerical range of any k real-valued linear functions on the n×p complex Stiefel manifold under the condition k2n2p+1. Revealing the hidden convexity of fractional linear programming on the complex Stiefel manifold, a first-time study, serves as an impactful application.
{"title":"Numerical range of real-valued linear mapping on the complex Stiefel manifold: Convexity and application","authors":"Hanzhi Chen,&nbsp;Zhenhong Huang,&nbsp;Mengmeng Song,&nbsp;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}
引用次数: 0
Polynomial codimension growth of superalgebras with superautomorphism
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.035
Sara Accomando
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.
{"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}
引用次数: 0
Constructing adjacency and distance cospectral graphs via regular rational orthogonal matrix
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.033
Lihuan Mao , Yuanhang Xu , Fenjin Liu , Bei Liu
Two graphs G and H are cospectral if they share the same spectrum. Constructing cospectral 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.
{"title":"Constructing adjacency and distance cospectral graphs via regular rational orthogonal matrix","authors":"Lihuan Mao ,&nbsp;Yuanhang Xu ,&nbsp;Fenjin Liu ,&nbsp;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}
引用次数: 0
The Rao-Mitra-Bhimasankaram relation is strongly antisymmetric
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-27 DOI: 10.1016/j.laa.2025.01.029
Oskar Maria Baksalary , Dennis Bernstein
For n×m complex matrices A and B, the Rao-Mitra-Bhimasankaram (RMB) relation RMB, defined by ARMBB if AAA=ABA, is reflexive and antisymmetric, but not transitive. This paper shows that, despite the lack of transitivity, RMB is strongly antisymmetric in the sense that, for all integers n2, A1RMBRMBAnRMBA1 implies A1==An. The proof of this result is based on a novel proof that RMB is antisymmetric.
{"title":"The Rao-Mitra-Bhimasankaram relation is strongly antisymmetric","authors":"Oskar Maria Baksalary ,&nbsp;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}
引用次数: 0
Rosenbrock's theorem on system matrices over elementary divisor domains
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.028
Froilán M. Dopico , Vanni Noferini , Ion Zaballa
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 G with the Smith form of an irreducible polynomial system matrix P giving rise to G and the Smith form of a submatrix of P. 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 P with entries in an arbitrary elementary divisor domain R and matrices G with entries in the field of fractions of R. 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 P itself has entries in the field of fractions of the elementary divisor domain.
{"title":"Rosenbrock's theorem on system matrices over elementary divisor domains","authors":"Froilán M. Dopico ,&nbsp;Vanni Noferini ,&nbsp;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}
引用次数: 0
When does subtracting a rank-one approximation decrease tensor rank?
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.025
Emil Horobeţ , Ettore Teixeira Turatti
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.
{"title":"When does subtracting a rank-one approximation decrease tensor rank?","authors":"Emil Horobeţ ,&nbsp;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}
引用次数: 0
On the nullity of middle graphs
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.030
Xinmei Yuan , Danyi Li , Weigen Yan
Let G be a connected graph, and let L(G) and M(G) be the line graph and middle graph of G. Gutman and Sciriha (On the nullity of line graphs of trees, Discrete Mathematics, 232 (2001), 35-45) proved that the nullity η(L(T)) of L(T) of a tree T satisfies η(L(T))=0 or η(L(T))=1. But the problem to determine which trees T satisfy η(L(T))=0 or η(L(T))=1 is still open. In this paper, we prove that η(M(G))=1 if G is a bipartite graph, and η(M(G))=0 otherwise. As an application, we show that η(G(n,m))=1 for the so-called silicate network G(n,m) obtained from the hexagonal lattice in the context of statistical physics.
{"title":"On the nullity of middle graphs","authors":"Xinmei Yuan ,&nbsp;Danyi Li ,&nbsp;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}
引用次数: 0
The dimension formula for certain twisted Jacquet modules of a cuspidal representation of GL(n,Fq)
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-22 DOI: 10.1016/j.laa.2025.01.027
Kumar Balasubramanian , Himanshi Khurana
Let n2 be a positive integer. Let F be the finite field of order q and G=GL(n,F). Let P=MN be the standard parabolic subgroup of G corresponding to the partition (k,nk). Let AM((nk)×k,F) be a rank t matrix. In this paper, we compute the dimension formula for the twisted Jacquet module πN,ψA that depends on n,k and t, when π is an irreducible cuspidal representation of G and ψA is a character of N associated with A.
{"title":"The dimension formula for certain twisted Jacquet modules of a cuspidal representation of GL(n,Fq)","authors":"Kumar Balasubramanian ,&nbsp;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}
引用次数: 0
Linear preservers of parallel matrix pairs with respect to the k-numerical radius
IF 1 3区 数学 Q1 MATHEMATICS Pub Date : 2025-01-21 DOI: 10.1016/j.laa.2025.01.019
Bojan Kuzma , Chi-Kwong Li , Edward Poon , Sushil Singla
Let 1k<n be integers. Two n×n matrices A and B form a parallel pair with respect to the k-numerical radius wk if wk(A+μB)=wk(A)+wk(B) for some scalar μ with |μ|=1; they form a TEA (triangle equality attaining) pair if the preceding equation holds for μ=1. We classify linear bijections on Mn and on Hn which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of wk-isometries, except for some exceptional maps on Hn when n=2k.
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Linear Algebra and its Applications
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