Journal of Algebra最新文献

{"title":"Local Hilbert–Schmidt stability","authors":"Francesco Fournier-Facio ,&nbsp;Maria Gerasimova ,&nbsp;Pieter Spaas","doi":"10.1016/j.jalgebra.2024.08.042","DOIUrl":"10.1016/j.jalgebra.2024.08.042","url":null,"abstract":"<div><div>We introduce a notion of local Hilbert–Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert–Schmidt stable but not Hilbert–Schmidt stable. For amenable groups, we provide a criterion for local Hilbert–Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert–Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert–Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert–Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445112","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":"Ranks of quadratic twists of Jacobians of generalized Mordell curves","authors":"Tomasz Jędrzejak","doi":"10.1016/j.jalgebra.2024.08.041","DOIUrl":"10.1016/j.jalgebra.2024.08.041","url":null,"abstract":"<div><div>Consider a two-parameter family of hyperelliptic curves <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>−</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> defined over <span><math><mi>Q</mi></math></span>, and their Jacobians <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> where <em>q</em> is an odd prime and without loss of generality <em>b</em> is a non-zero squarefree integer. The curve <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> is a quadratic twist by <em>b</em> of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> (a generalized Mordell curve of degree <em>q</em>). First, we obtain a few upper bounds for the ranks e.g., if the class number of <span><math><mi>Q</mi><mrow><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span> is odd, <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mrow><mo>(</mo><mi>mod</mi><mspace></mspace><mn>4</mn><mo>)</mo></mrow></math></span> and any prime divisor of <span><math><mspace></mspace><mn>2</mn><mi>b</mi></math></span> not equal to <em>q</em> is a primitive root modulo <em>q</em> then <span><math><mi>rank</mi><mspace></mspace><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></math></span>. Then we focus on <span><math><mi>q</mi><mo>=</mo><mn>5</mn></math></span> and get the best possible bound (by 1) or even the exact value of rank (0). In particular, we found infinitely many <em>b</em> with any number of prime factors such that <span><math><mi>rank</mi><mspace></mspace><msub><mrow><mi>J</mi></mrow><mrow><mn>5</mn><mo>,</mo><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></span>. We deduce as conclusions the complete list (or the bounds for the number) of rational points on <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn><mo>,</mo><mi>b</mi></mrow></msub></math></span> in such cases. Finally, we found for any given <em>q</em> infinitely many non-isomorphic curves <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> such that <span><math><mi>rank</mi><mspace></mspace><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444920","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":"Depth zero supercuspidal representations of classical groups into L-packets: The typically almost symmetric case","authors":"Geo Kam-Fai Tam","doi":"10.1016/j.jalgebra.2024.09.024","DOIUrl":"10.1016/j.jalgebra.2024.09.024","url":null,"abstract":"<div><div>We classify what we call “typically almost symmetric” depth zero supercuspidal representations of a classical group over a local field of odd residual characteristic into L-packets. Our main results resolve an ambiguity in the paper of Lust-Stevens <span><span>[24]</span></span> in this case, where they could only classify these representations in two or four, if not one, L-packets. By assuming the expected numbers of supercuspidal representations in the L-packets, we employ only simple properties of the representations to prove the main results. In particular, we do not require any deep calculations of character values. With the same method, we also compute the parity of a (conjugate-)self-dual depth zero supercuspidal representation of a general linear group.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444748","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":"Realization of Lie superalgebras G(3) and F(4) as symmetries of supergeometries","authors":"Boris Kruglikov,&nbsp;Andreu Llabrés","doi":"10.1016/j.jalgebra.2024.08.035","DOIUrl":"10.1016/j.jalgebra.2024.08.035","url":null,"abstract":"<div><div>For every parabolic subgroup <em>P</em> of a Lie supergroup <em>G</em> the homogeneous superspace <span><math><mi>G</mi><mo>/</mo><mi>P</mi></math></span> carries a <em>G</em>-invariant supergeometry. We address the problem whether <span><math><mi>g</mi><mo>=</mo><mrow><mi>Lie</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the maximal (local and global) symmetry of this supergeometry in the case of exceptional Lie superalgebras <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>F</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>. Our approach is to consider the negatively graded Lie superalgebras for every choice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, with reduction of the structure group when required (2 resp 3 special cases), thus realizing <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>F</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> as symmetries of supergeometries. This gives 19 inequivalent <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span>-supergeometries and 55 inequivalent <span><math><mi>F</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>-supergeometries, in majority of cases (17 resp 52 cases) those being encoded as vector superdistributions. We describe those supergeometries and realize supersymmetry explicitly.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529845","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":"On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties","authors":"C. Bagiński ,&nbsp;G. Gromadzki ,&nbsp;R.A. Hidalgo","doi":"10.1016/j.jalgebra.2024.09.012","DOIUrl":"10.1016/j.jalgebra.2024.09.012","url":null,"abstract":"<div><div>A continuous action of a finite group <em>G</em> on a closed orientable surface <em>X</em> is said to be gpnf (Gilman purely non-free) if every element of <em>G</em> has a fixed point on <em>X</em>. We prove that the biggest order <span><math><mi>μ</mi><mo>(</mo><mi>g</mi><mo>)</mo></math></span>, of a gpnf-action on a surface of even genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span>, is bounded below by 8<em>g</em> and that this bound is sharp for infinitely many even <em>g</em> as well. This provides, for even genera, a gpnf-action analog of the celebrated Accola-Maclachlan bound <span><math><mn>8</mn><mi>g</mi><mo>+</mo><mn>8</mn></math></span> for arbitrary finite continuous actions. We also describe the asymptotic behavior of <em>μ</em>. We define <span><math><mi>M</mi></math></span> as the set of values of the function <span><math><mover><mrow><mi>μ</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>g</mi><mo>)</mo><mo>=</mo><mi>μ</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>/</mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> and its subsets <span><math><msub><mrow><mi>M</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span> corresponding to even and odd genera <em>g</em>. We show that the set <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, of accumulation points of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, consists of a single number 8. If <em>g</em> is odd, then we prove that <span><math><mn>4</mn><mi>g</mi><mo>≤</mo><mi>μ</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>&lt;</mo><mn>8</mn><mi>g</mi></math></span>. We conjecture that this lower bound is sharp for infinitely many odd <em>g</em>. Finally, we prove that this conjecture implies that 4 is the only element of <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mo>−</mo></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, leading to <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>=</mo><mo>{</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>}</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444921","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 characteristic subgroups of an abelian p-group","authors":"K. Robin McLean","doi":"10.1016/j.jalgebra.2024.10.001","DOIUrl":"10.1016/j.jalgebra.2024.10.001","url":null,"abstract":"<div><div>We describe the characteristic subgroups of a transitive abelian 2-group, thus completing Kaplansky's study in section 18 of his classic monograph <em>Infinite abelian groups</em>. This appears to be a new result even for finite abelian 2-groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444917","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":"Fusion categories over non-algebraically closed fields","authors":"Sean Sanford","doi":"10.1016/j.jalgebra.2024.09.010","DOIUrl":"10.1016/j.jalgebra.2024.09.010","url":null,"abstract":"<div><div>Several complications arise when attempting to work with fusion categories over arbitrary fields. Here we describe some of the new phenomena that occur when the field is not algebraically closed, and we adapt tools such as the Frobenius-Perron dimension in order to accommodate these new effects.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427062","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":"Equivariant oriented homology of the affine Grassmannian","authors":"Changlong Zhong","doi":"10.1016/j.jalgebra.2024.09.009","DOIUrl":"10.1016/j.jalgebra.2024.09.009","url":null,"abstract":"<div><div>We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the Goresky-Kottwitz-MacPherson (GKM) condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the formal affine Demazure algebra.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427063","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":"Lifting elementary Abelian covers of curves","authors":"Jianing Yang","doi":"10.1016/j.jalgebra.2024.09.007","DOIUrl":"10.1016/j.jalgebra.2024.09.007","url":null,"abstract":"<div><div>Given a Galois cover of curves <em>f</em> over a field of characteristic <em>p</em>, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is <em>f</em>. In this paper, we consider the case where the Galois groups are elementary abelian <em>p</em>-groups. We prove a combinatorial criterion for lifting an elementary abelian <em>p</em>-cover, dependent on the branch loci of lifts of its <em>p</em>-cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, we analyze lifts for several families of <span><math><msup><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></math></span>-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427061","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":"Recognition and constructive membership for purely hyperbolic groups acting on trees","authors":"Ari Markowitz","doi":"10.1016/j.jalgebra.2024.09.008","DOIUrl":"10.1016/j.jalgebra.2024.09.008","url":null,"abstract":"<div><div>We present an algorithm which takes as input a finite set <em>X</em> of automorphisms of a simplicial tree, and outputs a generating set <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mo>〈</mo><mi>X</mi><mo>〉</mo></math></span> such that either <span><math><mo>〈</mo><mi>X</mi><mo>〉</mo></math></span> is purely hyperbolic and <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is a free basis of <span><math><mo>〈</mo><mi>X</mi><mo>〉</mo></math></span>, or <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> contains a non-trivial elliptic element. As a special case, the algorithm decides whether a finitely generated group acting on a locally finite tree is discrete and free. This algorithm, which is based on Nielsen's reduction method, works by repeatedly applying Nielsen transformations to <em>X</em> to minimise the generators of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> with respect to a given pre-well-ordering. We use this algorithm to solve the constructive membership problem for finitely generated purely hyperbolic automorphism groups of trees. We provide a <span>Magma</span> implementation of these algorithms, and report its performance.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437718","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}