Journal of Algebra最新文献

{"title":"Corrigendum to “The rotating normal form of braids is regular” [J. Algebra 501 (2018) 545–570]","authors":"Jean Fromentin","doi":"10.1016/j.jalgebra.2024.10.040","DOIUrl":"10.1016/j.jalgebra.2024.10.040","url":null,"abstract":"","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 253-254"},"PeriodicalIF":0.8,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720433","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}

{"title":"Hecke symmetries associated with twisted polynomial algebras in 3 indeterminates","authors":"Nikita Shishmarov,&nbsp;Serge Skryabin","doi":"10.1016/j.jalgebra.2024.11.012","DOIUrl":"10.1016/j.jalgebra.2024.11.012","url":null,"abstract":"<div><div>We consider Hecke symmetries on a 3-dimensional vector space with the associated <em>R</em>-symmetric algebra isomorphic to the polynomial algebra <span><math><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>]</mo></math></span> twisted by an automorphism. The main result states that any such a Hecke symmetry is itself a twist of a Hecke symmetry with the associated <em>R</em>-symmetric algebra isomorphic to <span><math><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>]</mo></math></span>. This allows us to describe equivalence classes of such Hecke symmetries.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 538-570"},"PeriodicalIF":0.8,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745178","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}

{"title":"Local cohomology tables of sequentially almost Cohen-Macaulay modules","authors":"Cheng Meng","doi":"10.1016/j.jalgebra.2024.10.049","DOIUrl":"10.1016/j.jalgebra.2024.10.049","url":null,"abstract":"<div><div>Let <em>R</em> be a polynomial ring over a field. We introduce the concept of sequentially almost Cohen-Macaulay modules and describe the extremal rays of the cone of local cohomology tables of finitely generated graded <em>R</em>-modules which are sequentially almost Cohen-Macaulay, and describe some cases when the local cohomology table of a module of dimension 3 has a nontrivial decomposition.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 596-627"},"PeriodicalIF":0.8,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142742871","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}

{"title":"Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras","authors":"Nobuyoshi Takahashi","doi":"10.1016/j.jalgebra.2024.11.010","DOIUrl":"10.1016/j.jalgebra.2024.11.010","url":null,"abstract":"<div><div>Let <em>T</em> be a Lie-Yamaguti algebra whose standard enveloping Lie algebra <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is semisimple and <span><math><mo>[</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>]</mo><mo>=</mo><mi>T</mi></math></span>. Then we give a description of representations of <em>T</em> in terms of representations of <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> with certain additional data. Similarly, if <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> is an infinitesimal <em>s</em>-manifold such that <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is semisimple, then any representation of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> comes from a representation of <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"664 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regular ring properties degraded through inverse limits","authors":"Pere Ara ,&nbsp;Ken Goodearl ,&nbsp;Kevin C. O'Meara ,&nbsp;Enrique Pardo ,&nbsp;Francesc Perera","doi":"10.1016/j.jalgebra.2024.11.006","DOIUrl":"10.1016/j.jalgebra.2024.11.006","url":null,"abstract":"<div><div>We give a number of constructions where inverse limits seriously degrade properties of regular rings, such as unit-regularity, diagonalisation of matrices, and finite stable rank. This raises the possibility of using inverse limits to answer the long standing Separativity Problem (in the negative).</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"664 ","pages":"Pages 365-397"},"PeriodicalIF":0.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parabolic Hitchin connection","authors":"Zakaria Ouaras","doi":"10.1016/j.jalgebra.2024.11.008","DOIUrl":"10.1016/j.jalgebra.2024.11.008","url":null,"abstract":"<div><div>In this paper, we present an algebro-geometric construction of the Hitchin connection in the parabolic setting for a fixed determinant line bundle. Our strategy is based on Hecke modifications, where we provide a decomposition formula for the parabolic determinant line bundle and the canonical line bundle of the moduli space of parabolic bundles. As a special case, we construct a Hitchin connection on the moduli space of vector bundles with fixed, not necessarily trivial, determinant.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 628-678"},"PeriodicalIF":0.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142742870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An extension of Schur's irreducibility result","authors":"Ankita Jindal ,&nbsp;Sudesh Kaur Khanduja","doi":"10.1016/j.jalgebra.2024.10.047","DOIUrl":"10.1016/j.jalgebra.2024.10.047","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; be an integer. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; belonging to &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be a monic polynomial which is irreducible modulo all primes less than or equal to &lt;em&gt;n&lt;/em&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be polynomials in &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; each having degree less than &lt;span&gt;&lt;math&gt;&lt;mi&gt;deg&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be an integer. Assume that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and the content of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are coprime with &lt;em&gt;n&lt;/em&gt;!. In the present paper, we prove that the polynomial &lt;span&gt;&lt;math&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; is irreducible over the field &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of rational numbers. This generalizes a well known result of Schur which states that the polynomial &lt;span&gt;&lt;math&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; is irreducible over &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; when each &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. The present paper also extends a result of Filaseta thereby leading to a generalization of the classical Schönemann Irreducibility Criterion.&lt;/div&gt;&lt;/d","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"664 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723069","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}

{"title":"The non-decreasing condition on g-vectors","authors":"Mohamad Haerizadeh,&nbsp;Siamak Yassemi","doi":"10.1016/j.jalgebra.2024.11.005","DOIUrl":"10.1016/j.jalgebra.2024.11.005","url":null,"abstract":"<div><div>The non-decreasing condition on g-vectors is introduced. Our study shows that this condition is both necessary and sufficient to ensure that the generically indecomposable direct summands of a given g-vector are linearly independent. Additionally, we prove that for any finite dimensional algebra Λ, under the non-decreasing condition, the number of generically indecomposable irreducible components that appear in the decomposition of a given generically <em>τ</em>-reduced component is lower than or equal to <span><math><mo>|</mo><mi>Λ</mi><mo>|</mo></math></span>. This solves the conjecture concerning the cardinality of component clusters by Cerulli-Labardini-Schröer, in a reasonable generality. Lastly, we study numerical criteria to check the wildness of g-vectors.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 571-595"},"PeriodicalIF":0.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745179","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}

{"title":"The commuting variety of pgln","authors":"Vlad Roman","doi":"10.1016/j.jalgebra.2024.10.034","DOIUrl":"10.1016/j.jalgebra.2024.10.034","url":null,"abstract":"<div><div>We are considering the commuting variety of the Lie algebra <span><math><msub><mrow><mi>pgl</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> over an algebraically closed field of characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span>, namely the set of pairs <span><math><mo>{</mo><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>pgl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>pgl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo><mo>=</mo><mn>0</mn><mo>}</mo></math></span>. We prove that if <span><math><mi>n</mi><mo>=</mo><mi>p</mi><mi>r</mi></math></span>, then there are precisely two irreducible components, of dimensions <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>n</mi><mo>−</mo><mn>2</mn></math></span>. We also prove that the variety <span><math><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>×</mo><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>|</mo><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo><mo>=</mo><mi>ζ</mi><mi>I</mi><mo>}</mo></math></span> is irreducible of dimension <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>n</mi><mo>/</mo><mi>d</mi></math></span>, where <em>ζ</em> is a root of unity of order <em>d</em> with <em>d</em> dividing <em>n</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 229-242"},"PeriodicalIF":0.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704946","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}

{"title":"On stable equivalences of Morita type and nilpotent blocks","authors":"Conghui Li","doi":"10.1016/j.jalgebra.2024.10.039","DOIUrl":"10.1016/j.jalgebra.2024.10.039","url":null,"abstract":"<div><div>In this note, we give a new proof by module-theoretic methods for a result of Puig asserting that blocks which are stable equivalent of Morita type to nilpotent blocks are also nilpotent.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 243-252"},"PeriodicalIF":0.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704947","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}