Journal of AlgebraPub Date : 2025-04-14DOI: 10.1016/j.jalgebra.2025.04.004
Michael K. Brown , Hailong Dao , Prashanth Sridhar
{"title":"Periodicity of ideals of minors in free resolutions","authors":"Michael K. Brown , Hailong Dao , Prashanth Sridhar","doi":"10.1016/j.jalgebra.2025.04.004","DOIUrl":"10.1016/j.jalgebra.2025.04.004","url":null,"abstract":"<div><div>We study the asymptotic behavior of the ideals of minors in minimal free resolutions over local rings. In particular, we prove that such ideals are eventually 2-periodic over complete intersections and Golod rings. We also establish general results on the stable behavior of ideals of minors in any infinite minimal free resolution. These ideals have intimate connections to trace ideals and cohomology annihilators. Constraints on the stable values attained by the ideals of minors in many situations are obtained, and they can be explicitly computed in certain cases.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 180-214"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854645","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}
Journal of AlgebraPub Date : 2025-04-14DOI: 10.1016/j.jalgebra.2025.03.052
Saudamini Nayak
{"title":"Weyl modules for queer Lie superalgebras","authors":"Saudamini Nayak","doi":"10.1016/j.jalgebra.2025.03.052","DOIUrl":"10.1016/j.jalgebra.2025.03.052","url":null,"abstract":"<div><div>We define global and local Weyl modules for <span><math><mi>q</mi><mo>⊗</mo><mi>A</mi></math></span>, where <span><math><mi>q</mi></math></span> is the queer Lie superalgebra and <em>A</em> is an associative commutative unital <span><math><mi>C</mi></math></span>-algebra. We prove that global Weyl modules are universal highest weight objects in certain category up to parity reversing functor Π. Then with the assumption that <em>A</em> is finitely generated, it is shown that the local Weyl modules are finite dimensional and further they are universal highest map-weight objects in certain category up to Π. Finally, we prove a tensor product property for local Weyl modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 196-223"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869706","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.045
J. Miquel Martínez
{"title":"Character degrees and local subgroups revisited","authors":"J. Miquel Martínez","doi":"10.1016/j.jalgebra.2025.03.045","DOIUrl":"10.1016/j.jalgebra.2025.03.045","url":null,"abstract":"<div><div>Let <em>p</em> and <em>q</em> be different primes and let <em>G</em> be a finite <em>q</em>-solvable group. We prove that <span><math><msub><mrow><mi>Irr</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>⊆</mo><msub><mrow><mi>Irr</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> if and only if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo><mo>⊆</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> for some <span><math><mi>P</mi><mo>∈</mo><msub><mrow><mi>Syl</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>Q</mi><mo>∈</mo><msub><mrow><mi>Syl</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Further, if <em>B</em> is a <em>q</em>-block of <em>G</em> and <em>p</em> does not divide the degree of any character in <span><math><mi>Irr</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span> then we prove that a Sylow <em>p</em>-subgroup of <em>G</em> is normalized by a defect group of <em>B</em>. This removes the <em>p</em>-solvability condition of two theorems of G. Navarro and T.R. Wolf.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 311-317"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143867721","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.050
Stephen Donkin
{"title":"Greene's Theorem and ideals of the group algebra of a symmetric group","authors":"Stephen Donkin","doi":"10.1016/j.jalgebra.2025.03.050","DOIUrl":"10.1016/j.jalgebra.2025.03.050","url":null,"abstract":"<div><div>We show that certain factor rings of the group algebra of a symmetric group have natural bases of group elements. These include the factor rings studied by Raghavan, Samuel and Subrahmanyam, <span><span>[19]</span></span> and by Doty, <span><span>[8]</span></span>. We also give generators for the annihilator of certain permutation modules for symmetric groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 159-179"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854727","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.038
Eugenio Giannelli , Niccolò Mecacci
{"title":"Even degree characters in blocks of symmetric and alternating groups","authors":"Eugenio Giannelli , Niccolò Mecacci","doi":"10.1016/j.jalgebra.2025.03.038","DOIUrl":"10.1016/j.jalgebra.2025.03.038","url":null,"abstract":"<div><div>We show that for every odd prime number <em>p</em>, every <em>p</em>-block of positive defect of a symmetric group admits irreducible characters of even degree. A similar result is obtained for alternating groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 237-251"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143858887","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.039
Valentina Pepe
{"title":"On subspaces defining linear sets of maximum rank","authors":"Valentina Pepe","doi":"10.1016/j.jalgebra.2025.03.039","DOIUrl":"10.1016/j.jalgebra.2025.03.039","url":null,"abstract":"<div><div>Let <em>V</em> denote an <em>r</em>-dimensional <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>-vector space. Let <em>U</em> and <em>W</em> be <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-subspaces of <em>V</em> and let <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>W</mi></mrow></msub></math></span> be the projective points of <span><math><mrow><mi>PG</mi></mrow><mspace></mspace><mo>(</mo><mi>V</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> defined by <em>U</em> and <em>W</em> respectively. We address the problem of when <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>W</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> under the hypothesis that <em>U</em> and <em>W</em> have maximum dimension, i.e., <span><math><msub><mrow><mi>dim</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub><mo></mo><mi>W</mi><mo>=</mo><msub><mrow><mi>dim</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub><mo></mo><mi>U</mi><mo>=</mo></math></span> <span><math><mi>r</mi><mi>n</mi><mo>−</mo><mi>n</mi></math></span>, and we give a complete characterization for <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 378-407"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859697","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.04.003
Ievgen Bondarenko
{"title":"The word problem and growth of groups","authors":"Ievgen Bondarenko","doi":"10.1016/j.jalgebra.2025.04.003","DOIUrl":"10.1016/j.jalgebra.2025.04.003","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> denote the word problem in a finitely generated group <em>G</em>. We consider the complexity of <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> with respect to standard deterministic Turing machines. Let <span><math><msub><mrow><mi>DTIME</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><msub><mrow><mi>DTIME</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> be the complexity classes of languages solved in time <span><math><mi>O</mi><mo>(</mo><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> by Turing machines with a single tape and multiple tapes respectively. We prove that <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> if and only if <em>G</em> is virtually nilpotent. We relate the complexity of the word problem and the growth of groups by showing that <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∉</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>o</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>γ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>γ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the growth function of <em>G</em>. We prove that <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for strongly contracting automaton groups, <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> for groups generated by bounded automata, and <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>n</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> for groups generated by polynomial automata. In particular, for the Grigorchuk group, <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∉</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1.7674</mn></mrow></msup><mo>)</mo></math></span> and <span><math><msub><mrow><mi>WP</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>DTIME</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 252-266"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143858889","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.044
Hengfei Lu
{"title":"Remark on the distributions supported on the nilpotent cone","authors":"Hengfei Lu","doi":"10.1016/j.jalgebra.2025.03.044","DOIUrl":"10.1016/j.jalgebra.2025.03.044","url":null,"abstract":"<div><div>Let <em>F</em> be a local field of characteristic zero. This note gives two examples for the vanishing of certain distributions supported on the nilpotent cone. Then we give a shorter proof to the multiplicity one theorem for the symmetric variety <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math></span> where <em>D</em> is a 4-dimensional quaternion algebra over <em>F</em> and <em>E</em> is a quadratic field extension of <em>F</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 139-158"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854726","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.046
Omar Dennaoui, Jonathon Villareal
{"title":"Contractibility of the orbit space of a saturated fusion system after Steinberg","authors":"Omar Dennaoui, Jonathon Villareal","doi":"10.1016/j.jalgebra.2025.03.046","DOIUrl":"10.1016/j.jalgebra.2025.03.046","url":null,"abstract":"<div><div>Recently, Steinberg used discrete Morse theory to give a new proof of a theorem of Symonds that the orbit space of the poset of nontrivial <em>p</em>-subgroups of a finite group is contractible. We extend Steinberg's argument in two ways, covering more general versions of the theorem that were already known. In particular, following a strategy of Libman, we give a discrete Morse theoretic argument for the contractibility of the orbit space of a saturated fusion system.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 267-277"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864659","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}
Journal of AlgebraPub Date : 2025-04-11DOI: 10.1016/j.jalgebra.2025.03.040
Francis Brown
{"title":"Multivariable Vandermonde determinants, amalgams of matrices and Specht modules","authors":"Francis Brown","doi":"10.1016/j.jalgebra.2025.03.040","DOIUrl":"10.1016/j.jalgebra.2025.03.040","url":null,"abstract":"<div><div>Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of a matrix which is obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 253-278"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870492","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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