Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.10.052
Naihuan Jing , Yu Wu
In this paper, we use vertex operator techniques to compute character values on unipotent classes of . By realizing the Grothendieck ring as Fock spaces, we formulate the Murnaghan-Nakayama rule of between Schur functions colored by an orbit ϕ of linear characters of and another orbit of modified Hall-Littlewood functions colored by under the Frobenius automorphisms. Our formulation of character values using vertex operators offers a practical approach for computing special values at unipotent classes for . As an application, these vertex-algebraic techniques allow us to derive the Steinberg characters of , results that were previously obtained by Curtis, Lehrer, and Tits through the geometry of homology groups of spherical buildings, and by Springer and Zelevinsky via the theory of Hopf algebras.
{"title":"On character values of GLn(Fq)","authors":"Naihuan Jing , Yu Wu","doi":"10.1016/j.jalgebra.2025.10.052","DOIUrl":"10.1016/j.jalgebra.2025.10.052","url":null,"abstract":"<div><div>In this paper, we use vertex operator techniques to compute character values on unipotent classes of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>. By realizing the Grothendieck ring <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>⨁</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mi>R</mi><mo>(</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> as Fock spaces, we formulate the Murnaghan-Nakayama rule of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> between Schur functions colored by an orbit <em>ϕ</em> of linear characters of <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></math></span> and another orbit of modified Hall-Littlewood functions colored by <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>t</mi><mo>−</mo><mn>1</mn></math></span> under the Frobenius automorphisms. Our formulation of character values using vertex operators offers a practical approach for computing special values at unipotent classes for <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>. As an application, these vertex-algebraic techniques allow us to derive the Steinberg characters of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>, results that were previously obtained by Curtis, Lehrer, and Tits through the geometry of homology groups of spherical buildings, and by Springer and Zelevinsky via the theory of Hopf algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 214-229"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.008
Sebastian Debus , Tobias Metzlaff
The action of a Weyl group on the associated weight lattice induces an additive action on the symmetric algebra and a multiplicative action on the group algebra of the lattice. We show that the coinvariant space of the multiplicative action affords the regular representation and is isomorphic to a space of multiplicative harmonics, which corresponds to existing results for additive coinvariants of reflection groups. We then design an algorithm to compute a multiplicative coinvariant basis from an additive one. The algorithm preserves isotypic decomposition and graded structure and enables the study of multiplicative coinvariants by integrating combinatorial knowledge from the additive setting. We investigate the Weyl groups of type A and C to find new explicit equivariant maps and combinatorial structure.
{"title":"Additive and multiplicative coinvariant spaces of Weyl groups in the light of harmonics and graded transfer","authors":"Sebastian Debus , Tobias Metzlaff","doi":"10.1016/j.jalgebra.2025.11.008","DOIUrl":"10.1016/j.jalgebra.2025.11.008","url":null,"abstract":"<div><div>The action of a Weyl group on the associated weight lattice induces an additive action on the symmetric algebra and a multiplicative action on the group algebra of the lattice. We show that the coinvariant space of the multiplicative action affords the regular representation and is isomorphic to a space of multiplicative harmonics, which corresponds to existing results for additive coinvariants of reflection groups. We then design an algorithm to compute a multiplicative coinvariant basis from an additive one. The algorithm preserves isotypic decomposition and graded structure and enables the study of multiplicative coinvariants by integrating combinatorial knowledge from the additive setting. We investigate the Weyl groups of type A and C to find new explicit equivariant maps and combinatorial structure.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"690 ","pages":"Pages 806-831"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145614725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.006
James East , Matthias Fresacher , P.A. Azeef Muhammed , Timothy Stokes
DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular ⁎-, and ⁎-regular semigroups; concrete examples include diagram monoids, linear monoids, relation monoids, among many others. In this paper we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular ⁎-semigroups of the first and third authors. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.
dc -半群用域和值域运算对关联系统进行建模，并包含了许多重要的类，如逆半群、限制半群、Ehresmann半群、正则半群和正则半群；具体的例子包括图一元群、线性一元群、关系一元群等。本文证明了dc -半群的范畴与某些双序范畴的范畴同构，这些双序范畴的对象集是Jones意义上的投影代数。这将最近的类群方法扩展到第一和第三作者的正则半群。通过构造投影代数范畴中遗忘函子的左伴随，证明了自由dc -半群的存在性。
{"title":"Categorical representation of DRC-semigroups","authors":"James East , Matthias Fresacher , P.A. Azeef Muhammed , Timothy Stokes","doi":"10.1016/j.jalgebra.2025.11.006","DOIUrl":"10.1016/j.jalgebra.2025.11.006","url":null,"abstract":"<div><div>DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular ⁎-, and ⁎-regular semigroups; concrete examples include diagram monoids, linear monoids, relation monoids, among many others. In this paper we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular ⁎-semigroups of the first and third authors. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 230-291"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.016
Yu Zhang , Xiaomin Tang
Let denote the quantum coordinate ring of the space of skew-symmetric matrices where . We show that admits the structure of a symmetric CGL-extension. Leveraging this finding, we extend the construction of quantum cluster algebras through symmetric CGL-extensions under additional conditions. Consequently, we obtain an explicit quantum cluster structure on . The cornerstone of our approach lies in utilizing the exchange matrix from the quantum coordinate ring of the unipotent subgroup in a symmetric Kac–Moody group G, which is associated with a particular Weyl group element. In this work, by using the different method we generalize existing results in [22], originally established for , to the case of arbitrary positive integers .
{"title":"Cluster structure on the quantum coordinate ring of skew-symmetric matrices in general case","authors":"Yu Zhang , Xiaomin Tang","doi":"10.1016/j.jalgebra.2025.11.016","DOIUrl":"10.1016/j.jalgebra.2025.11.016","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> denote the quantum coordinate ring of the space of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> skew-symmetric matrices where <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. We show that <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> admits the structure of a symmetric CGL-extension. Leveraging this finding, we extend the construction of quantum cluster algebras through symmetric CGL-extensions under additional conditions. Consequently, we obtain an explicit quantum cluster structure on <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>. The cornerstone of our approach lies in utilizing the exchange matrix from the quantum coordinate ring of the unipotent subgroup <span><math><mi>N</mi><mo>(</mo><mi>w</mi><mo>)</mo></math></span> in a symmetric Kac–Moody group <em>G</em>, which is associated with a particular Weyl group element. In this work, by using the different method we generalize existing results in <span><span>[22]</span></span>, originally established for <span><math><mi>n</mi><mo>=</mo><mn>5</mn></math></span>, to the case of arbitrary positive integers <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"690 ","pages":"Pages 677-700"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145615637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.007
Hongju Zhao, Qiang Mu
We study toroidal vertex algebras and their modules over a general field of prime characteristic, and provide a conceptual construction of modular toroidal vertex algebras and their modules. As an example, we consider the toroidal vertex algebra associated with a toroidal Lie algebra and further construct a family of its quotients.
{"title":"Modular toroidal vertex algebras and their modules","authors":"Hongju Zhao, Qiang Mu","doi":"10.1016/j.jalgebra.2025.11.007","DOIUrl":"10.1016/j.jalgebra.2025.11.007","url":null,"abstract":"<div><div>We study toroidal vertex algebras and their modules over a general field of prime characteristic, and provide a conceptual construction of modular toroidal vertex algebras and their modules. As an example, we consider the toroidal vertex algebra associated with a toroidal Lie algebra and further construct a family of its quotients.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 88-127"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.10.053
Ishan Banerjee , Peter Huxford
We prove for and that the level m congruence subgroup of the braid group associated to the integral Burau representation is generated by mth powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle–Margalit, Nakamura, Stylianakis and Wajnryb.
{"title":"Generators for the level m congruence subgroups of braid groups","authors":"Ishan Banerjee , Peter Huxford","doi":"10.1016/j.jalgebra.2025.10.053","DOIUrl":"10.1016/j.jalgebra.2025.10.053","url":null,"abstract":"<div><div>We prove for <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span> that the level <em>m</em> congruence subgroup <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>[</mo><mi>m</mi><mo>]</mo></math></span> of the braid group <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> associated to the integral Burau representation <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> is generated by <em>m</em>th powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle–Margalit, Nakamura, Stylianakis and Wajnryb.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 1-16"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.10.055
D. Zhangazinova , A. Naurazbekova , U. Umirbaev
Let be an n-dimensional algebra with zero multiplication over a field K of characteristic 0. Then its universal (multiplicative) enveloping algebra in the variety of left-symmetric algebras is a homogeneous quadratic algebra generated by 2n elements , which contains both the polynomial algebra and the free associative algebra . We show that the automorphism groups of the polynomial algebra and the algebra are isomorphic for all , based on a detailed analysis of locally nilpotent derivations. In contrast, we show that this isomorphism does not hold for , and we provide a complete description of all automorphisms and locally nilpotent derivations of .
{"title":"Automorphisms and derivations of a universal left-symmetric enveloping algebra","authors":"D. Zhangazinova , A. Naurazbekova , U. Umirbaev","doi":"10.1016/j.jalgebra.2025.10.055","DOIUrl":"10.1016/j.jalgebra.2025.10.055","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be an <em>n</em>-dimensional algebra with zero multiplication over a field <em>K</em> of characteristic 0. Then its universal (multiplicative) enveloping algebra <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in the variety of left-symmetric algebras is a homogeneous quadratic algebra generated by 2<em>n</em> elements <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which contains both the polynomial algebra <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>K</mi><mo>[</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> and the free associative algebra <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>K</mi><mo>〈</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〉</mo></math></span>. We show that the automorphism groups of the polynomial algebra <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the algebra <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are isomorphic for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, based on a detailed analysis of locally nilpotent derivations. In contrast, we show that this isomorphism does not hold for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>, and we provide a complete description of all automorphisms and locally nilpotent derivations of <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"690 ","pages":"Pages 701-729"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145615579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.010
Victor H. Jorge-Pérez, Paulo Martins, Victor D. Mendoza-Rubio
The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring R and finitely generated R-modules M and N, we provide conditions involving quasi-homological dimensions where the equality , which we call Ischebeck's formula, holds. One of the results in this direction generalizes a well-known result of Ischebeck concerning modules of finite injective dimension, considering the quasi-injective dimension. On the other hand, we establish an inequality relating the quasi-projective dimension of a finitely generated module to its grade and introduce the concept of a quasi-perfect module as a natural generalization of a perfect module. We prove some results for this new concept similar to the classical results. Additionally, we provide a formula for the grade of finitely generated modules with finite quasi-injective dimension over a local ring, as well as grade inequalities for modules of finite quasi-projective dimension. In our study, Cohen-Macaulayness criteria are also obtained.
{"title":"Ischebeck's formula, grade and quasi-homological dimensions","authors":"Victor H. Jorge-Pérez, Paulo Martins, Victor D. Mendoza-Rubio","doi":"10.1016/j.jalgebra.2025.11.010","DOIUrl":"10.1016/j.jalgebra.2025.11.010","url":null,"abstract":"<div><div>The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring <em>R</em> and finitely generated <em>R</em>-modules <em>M</em> and <em>N</em>, we provide conditions involving quasi-homological dimensions where the equality <span><math><mi>sup</mi><mo></mo><mo>{</mo><mi>i</mi><mo>≥</mo><mn>0</mn><mo>:</mo><msubsup><mrow><mi>Ext</mi></mrow><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo>)</mo><mo>≠</mo><mn>0</mn><mo>}</mo><mo>=</mo><mi>depth</mi><mspace></mspace><mi>R</mi><mo>−</mo><mi>depth</mi><mspace></mspace><mi>M</mi></math></span>, which we call Ischebeck's formula, holds. One of the results in this direction generalizes a well-known result of Ischebeck concerning modules of finite injective dimension, considering the quasi-injective dimension. On the other hand, we establish an inequality relating the quasi-projective dimension of a finitely generated module to its grade and introduce the concept of a quasi-perfect module as a natural generalization of a perfect module. We prove some results for this new concept similar to the classical results. Additionally, we provide a formula for the grade of finitely generated modules with finite quasi-injective dimension over a local ring, as well as grade inequalities for modules of finite quasi-projective dimension. In our study, Cohen-Macaulayness criteria are also obtained.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"690 ","pages":"Pages 653-676"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145615577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.011
Fei Hu
We show that the eigenvalues of any polarized endomorphism acting on the ℓ-adic étale cohomology of a smooth projective variety satisfy certain parity and symmetry properties, as predicted by the standard conjectures. These properties were previously known for Frobenius endomorphisms. Besides the hard Lefschetz theorem, a key new ingredient is a recent Weil's Riemann hypothesis-type result due to J. Xie. We also prove a “Newton over Hodge” type property for abelian varieties and Grassmannians.
{"title":"Parity and symmetry of polarized endomorphisms on cohomology","authors":"Fei Hu","doi":"10.1016/j.jalgebra.2025.11.011","DOIUrl":"10.1016/j.jalgebra.2025.11.011","url":null,"abstract":"<div><div>We show that the eigenvalues of any polarized endomorphism acting on the <em>ℓ</em>-adic étale cohomology of a smooth projective variety satisfy certain parity and symmetry properties, as predicted by the standard conjectures. These properties were previously known for Frobenius endomorphisms. Besides the hard Lefschetz theorem, a key new ingredient is a recent Weil's Riemann hypothesis-type result due to J. Xie. We also prove a “Newton over Hodge” type property for abelian varieties and Grassmannians.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"690 ","pages":"Pages 793-805"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145615581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.017
Alessandro Contu
In 2021, Kashiwara–Kim–Oh–Park constructed cluster algebra structures on the Grothendieck rings of certain monoidal subcategories of the category of finite-dimensional representations of a quantum loop algebra, generalizing Hernandez–Leclerc's pioneering work from 2010. They stated the problem of finding explicit quivers for the seeds they used. We provide a solution by using Palu's generalized mutation rule applied to the cluster categories associated with certain algebras of global dimension at most 2, for example tensor products of path algebras of representation-finite quivers. Thus, our method is based on (and contributes to) the bridge, provided by cluster combinatorics, between the representation theory of quantum groups and that of quivers with relations.
{"title":"Solution of a problem in monoidal categorification by additive categorification","authors":"Alessandro Contu","doi":"10.1016/j.jalgebra.2025.11.017","DOIUrl":"10.1016/j.jalgebra.2025.11.017","url":null,"abstract":"<div><div>In 2021, Kashiwara–Kim–Oh–Park constructed cluster algebra structures on the Grothendieck rings of certain monoidal subcategories of the category of finite-dimensional representations of a quantum loop algebra, generalizing Hernandez–Leclerc's pioneering work from 2010. They stated the problem of finding explicit quivers for the seeds they used. We provide a solution by using Palu's generalized mutation rule applied to the cluster categories associated with certain algebras of global dimension at most 2, for example tensor products of path algebras of representation-finite quivers. Thus, our method is based on (and contributes to) the bridge, provided by cluster combinatorics, between the representation theory of quantum groups and that of quivers with relations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 128-185"},"PeriodicalIF":0.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号