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On the vanishing of the hyperdeterminant under certain symmetry conditions
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-12-02 DOI: 10.1016/j.jalgebra.2024.11.018
Enrique Arrondo , Alicia Tocino
{"title":"On the vanishing of the hyperdeterminant under certain symmetry conditions","authors":"Enrique Arrondo ,&nbsp;Alicia Tocino","doi":"10.1016/j.jalgebra.2024.11.018","DOIUrl":"10.1016/j.jalgebra.2024.11.018","url":null,"abstract":"<div><div>Given a vector space <em>V</em> over a field <span><math><mi>K</mi></math></span> whose characteristic is coprime with <em>d</em>!, let us decompose the vector space of multilinear forms <span><math><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⊗</mo><mover><mo>…</mo><mrow><mtext>(</mtext><mi>d</mi><mo>)</mo></mrow></mover><mo>⊗</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>λ</mi></mrow></msub><msub><mrow><mi>W</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>K</mi><mo>)</mo></math></span> according to the different partitions <em>λ</em> of <em>d</em>, i.e. the different representations of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>. In this paper we first give a decomposition <span><math><msub><mrow><mi>W</mi></mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>(</mo><mi>V</mi><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>⨁</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msubsup><mrow><mi>W</mi></mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>i</mi></mrow></msubsup><mo>(</mo><mi>V</mi><mo>,</mo><mi>K</mi><mo>)</mo></math></span>. We finally prove the vanishing of the hyperdeterminant of any <span><math><mi>F</mi><mo>∈</mo><mo>(</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>λ</mi><mo>≠</mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>)</mo><mo>⊕</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>i</mi></mrow></msubsup><mo>(</mo><mi>V</mi><mo>,</mo><mi>K</mi><mo>)</mo></math></span>. This improves the result in <span><span>[10]</span></span> and <span><span>[1]</span></span>, where the same result was proved without this new last summand.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 269-278"},"PeriodicalIF":0.8,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152714","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}
引用次数: 0
k summands of syzygies over rings of positive Burch index via canonical resolutions
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-29 DOI: 10.1016/j.jalgebra.2024.11.013
Michael DeBellevue, Claudia Miller
{"title":"k summands of syzygies over rings of positive Burch index via canonical resolutions","authors":"Michael DeBellevue,&nbsp;Claudia Miller","doi":"10.1016/j.jalgebra.2024.11.013","DOIUrl":"10.1016/j.jalgebra.2024.11.013","url":null,"abstract":"<div><div>In recent work, Dao and Eisenbud define the notion of a Burch index, expanding the notion of Burch rings of Dao, Kobayashi, and Takahashi, and show that for any module over a ring of Burch index at least 2, its <em>n</em>th syzygy contains direct summands of the residue field for <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span> or 5 and all <span><math><mi>n</mi><mo>≥</mo><mn>7</mn></math></span>. We investigate how this behavior is explained by the bar resolution formed from appropriate differential graded (dg) resolutions, yielding a new proof that includes all <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, which is sharp. When the module is Golod, we use instead the bar resolution formed from <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> resolutions to identify such <em>k</em> summands explicitly for all <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span> and show that the number of these grows exponentially as the homological degree increases.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 657-672"},"PeriodicalIF":0.8,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152660","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}
引用次数: 0
The first and second derivatives of the q-Rationals
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-28 DOI: 10.1016/j.jalgebra.2024.10.033
Justin Lasker
{"title":"The first and second derivatives of the q-Rationals","authors":"Justin Lasker","doi":"10.1016/j.jalgebra.2024.10.033","DOIUrl":"10.1016/j.jalgebra.2024.10.033","url":null,"abstract":"<div><div>The <em>q</em>-rationals, introduced by Valentin Ovsienko and Sophie Morier Genoud, are an extension of Gauss' <em>q</em>-integers. Like the <em>q</em>-integers, the <em>q</em>-rationals reduce to their non-quantized values at <span><math><mi>q</mi><mo>=</mo><mn>1</mn></math></span>. In this paper, I prove closed-form expressions for the first and second derivatives of the <em>q</em>-rationals at this point. My expressions are written in terms of the <em>q</em>-rationals' non-quantized values; both feature Thomae's function, whereas my expression for the second derivative additionally features a generalized form of the Dedekind sum.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 733-776"},"PeriodicalIF":0.8,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152723","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}
引用次数: 0
Corrigendum to “The rotating normal form of braids is regular” [J. Algebra 501 (2018) 545–570] 对 "辫子的旋转正则形式是正则的 "的更正 [J. Algebra 501 (2018) 545-570]
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-26 DOI: 10.1016/j.jalgebra.2024.10.040
Jean Fromentin
{"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}
引用次数: 0
Topology and monoid representations II: Left regular bands of groups and Hsiao's monoid of ordered G-partitions
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-22 DOI: 10.1016/j.jalgebra.2024.10.041
Benjamin Steinberg
{"title":"Topology and monoid representations II: Left regular bands of groups and Hsiao's monoid of ordered G-partitions","authors":"Benjamin Steinberg","doi":"10.1016/j.jalgebra.2024.10.041","DOIUrl":"10.1016/j.jalgebra.2024.10.041","url":null,"abstract":"<div><div>The goal of this paper is to use topological methods to compute Ext between irreducible representations of von Neumann regular monoids in which Green's <span><math><mi>L</mi></math></span>- and <span><math><mi>J</mi></math></span>-relations coincide (e.g., left regular bands). Our results subsume those of Margolis et al. (2015) <span><span>[22]</span></span>.</div><div>Applications include computing Ext between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered <em>G</em>-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product <span><math><mi>G</mi><mo>≀</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. More generally, these results work for CW left regular bands of abelian groups. These results generalize the results of Margolis et al. (2021) <span><span>[23]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"665 ","pages":"Pages 679-710"},"PeriodicalIF":0.8,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161296","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}
引用次数: 0
Hecke symmetries associated with twisted polynomial algebras in 3 indeterminates 3不定数中与扭曲多项式代数相关的Hecke对称性
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-22 DOI: 10.1016/j.jalgebra.2024.11.012
Nikita Shishmarov, Serge Skryabin
{"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}
引用次数: 0
Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras 具有半简单包络列的山口列代数的表征
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-20 DOI: 10.1016/j.jalgebra.2024.11.010
Nobuyoshi Takahashi
{"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}
引用次数: 0
Local cohomology tables of sequentially almost Cohen-Macaulay modules 序列概Cohen-Macaulay模的局部上同表
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-20 DOI: 10.1016/j.jalgebra.2024.10.049
Cheng Meng
{"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}
引用次数: 0
Regular ring properties degraded through inverse limits 通过反向极限退化的正则环特性
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.11.006
Pere Ara , Ken Goodearl , Kevin C. O'Meara , Enrique Pardo , Francesc Perera
{"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}
引用次数: 0
Parabolic Hitchin connection 抛物线连接
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.11.008
Zakaria Ouaras
{"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}
引用次数: 0
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