Pub Date : 2024-12-03DOI: 10.1016/j.jalgebra.2024.11.019
Laurence Barker
A pointed p-group is a pointed group such that P is a p-group. We parameterize the pointed p-groups on a group algebra or on a block algebra of a group algebra. This is equivalent to a parameterization of the isomorphism classes of indecomposable direct summands of the algebra as a bimodule with the group acting on the left and a p-subgroup acting on the right. The parameterization involves p-subgroups and irreducible characters of centralizers of p-subgroups.
{"title":"The pointed p-groups on a block algebra","authors":"Laurence Barker","doi":"10.1016/j.jalgebra.2024.11.019","DOIUrl":"10.1016/j.jalgebra.2024.11.019","url":null,"abstract":"<div><div>A pointed <em>p</em>-group is a pointed group <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>γ</mi></mrow></msub></math></span> such that <em>P</em> is a <em>p</em>-group. We parameterize the pointed <em>p</em>-groups on a group algebra or on a block algebra of a group algebra. This is equivalent to a parameterization of the isomorphism classes of indecomposable direct summands of the algebra as a bimodule with the group acting on the left and a <em>p</em>-subgroup acting on the right. The parameterization involves <em>p</em>-subgroups and irreducible characters of centralizers of <em>p</em>-subgroups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 251-268"},"PeriodicalIF":0.8,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152715","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 : 2024-12-03DOI: 10.1016/j.jalgebra.2024.11.017
Fulvio Gesmundo , Leonie Kayser , Simon Telen
A chopped ideal is obtained from a homogeneous ideal by considering only the generators of a fixed degree. We investigate cases in which the chopped ideal defines the same finite set of points as the original one-dimensional ideal. The complexity of computing these points from the chopped ideal is governed by the Hilbert function and regularity. We conjecture values for these invariants and prove them in many cases. We show that our conjecture is of practical relevance for symmetric tensor decomposition.
{"title":"Hilbert functions of chopped ideals","authors":"Fulvio Gesmundo , Leonie Kayser , Simon Telen","doi":"10.1016/j.jalgebra.2024.11.017","DOIUrl":"10.1016/j.jalgebra.2024.11.017","url":null,"abstract":"<div><div>A chopped ideal is obtained from a homogeneous ideal by considering only the generators of a fixed degree. We investigate cases in which the chopped ideal defines the same finite set of points as the original one-dimensional ideal. The complexity of computing these points from the chopped ideal is governed by the Hilbert function and regularity. We conjecture values for these invariants and prove them in many cases. We show that our conjecture is of practical relevance for symmetric tensor decomposition.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 415-445"},"PeriodicalIF":0.8,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152731","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}
Pub Date : 2024-12-02DOI: 10.1016/j.jalgebra.2024.11.018
Enrique Arrondo , Alicia Tocino
Given a vector space V over a field whose characteristic is coprime with d!, let us decompose the vector space of multilinear forms according to the different partitions λ of d, i.e. the different representations of . In this paper we first give a decomposition . We finally prove the vanishing of the hyperdeterminant of any . This improves the result in [10] and [1], where the same result was proved without this new last summand.
{"title":"On the vanishing of the hyperdeterminant under certain symmetry conditions","authors":"Enrique Arrondo , 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}
Pub Date : 2024-11-29DOI: 10.1016/j.jalgebra.2024.11.013
Michael DeBellevue, Claudia Miller
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 nth syzygy contains direct summands of the residue field for or 5 and all . 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 , which is sharp. When the module is Golod, we use instead the bar resolution formed from resolutions to identify such k summands explicitly for all and show that the number of these grows exponentially as the homological degree increases.
{"title":"k summands of syzygies over rings of positive Burch index via canonical resolutions","authors":"Michael DeBellevue, 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}
Pub Date : 2024-11-28DOI: 10.1016/j.jalgebra.2024.10.033
Justin Lasker
The q-rationals, introduced by Valentin Ovsienko and Sophie Morier Genoud, are an extension of Gauss' q-integers. Like the q-integers, the q-rationals reduce to their non-quantized values at . In this paper, I prove closed-form expressions for the first and second derivatives of the q-rationals at this point. My expressions are written in terms of the q-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.
{"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}
Pub Date : 2024-11-26DOI: 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}
Pub Date : 2024-11-22DOI: 10.1016/j.jalgebra.2024.10.041
Benjamin Steinberg
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 - and -relations coincide (e.g., left regular bands). Our results subsume those of Margolis et al. (2015) [22].
Applications include computing Ext between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered G-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product ). 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) [23].
{"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}
Pub Date : 2024-11-22DOI: 10.1016/j.jalgebra.2024.11.012
Nikita Shishmarov, Serge Skryabin
We consider Hecke symmetries on a 3-dimensional vector space with the associated R-symmetric algebra isomorphic to the polynomial algebra 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 R-symmetric algebra isomorphic to . This allows us to describe equivalence classes of such Hecke symmetries.
{"title":"Hecke symmetries associated with twisted polynomial algebras in 3 indeterminates","authors":"Nikita Shishmarov, 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}
Pub Date : 2024-11-20DOI: 10.1016/j.jalgebra.2024.11.010
Nobuyoshi Takahashi
Let T be a Lie-Yamaguti algebra whose standard enveloping Lie algebra is semisimple and . Then we give a description of representations of T in terms of representations of with certain additional data. Similarly, if is an infinitesimal s-manifold such that is semisimple, then any representation of comes from a representation of .
设 T 是一个标准包络李代数 L(T) 为半简单且 [T,T,T]=T的Lie-Yamaguti 代数。然后，我们用 L(T) 的表示来描述 T 的表示，并给出某些附加数据。同样，如果（T,σ）是一个无穷小 s-manifold，且 L(T) 是半简单的，那么（T,σ）的任何表示都来自 L(T) 的表示。
{"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}
Pub Date : 2024-11-20DOI: 10.1016/j.jalgebra.2024.10.049
Cheng Meng
Let R 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 R-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.
{"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}
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