Journal of Algebra最新文献

{"title":"Endomorphisms of Artin groups of type A˜n","authors":"Luis Paris ,&nbsp;Ignat Soroko","doi":"10.1016/j.jalgebra.2025.04.001","DOIUrl":"10.1016/j.jalgebra.2025.04.001","url":null,"abstract":"<div><div>We determine a classification of the endomorphisms of the Artin group of affine type <span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>⩾</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 809-830"},"PeriodicalIF":0.8,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106850","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":"Uniform exponential growth for groups with proper product actions on hyperbolic spaces","authors":"Renxing Wan ,&nbsp;Wenyuan Yang","doi":"10.1016/j.jalgebra.2025.03.028","DOIUrl":"10.1016/j.jalgebra.2025.03.028","url":null,"abstract":"<div><div>This paper studies the locally uniform exponential growth and product set growth for a finitely generated group <em>G</em> acting properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing property on factors, we prove that any finitely generated non-virtually abelian subgroup has uniform exponential growth. These assumptions are fulfilled in many hierarchically hyperbolic groups, including mapping class groups, specially cubulated groups and BMW groups.</div><div>Moreover, if <em>G</em> acts weakly acylindrically on each factor, we show that, with two exceptional classes of subgroups, <em>G</em> has uniform product set growth. As corollaries, this gives a complete classification of subgroups with product set growth for any group acting discretely on a simply connected manifold with pinched negative curvature, for groups acting acylindrically on trees, and for 3-manifold groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 189-235"},"PeriodicalIF":0.8,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825653","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":"A note on the non-existence of functors","authors":"E. Dror Farjoun ,&nbsp;S.O. Ivanov ,&nbsp;A. Krasilnikov ,&nbsp;A. Zaikovskii","doi":"10.1016/j.jalgebra.2025.03.037","DOIUrl":"10.1016/j.jalgebra.2025.03.037","url":null,"abstract":"<div><div>We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. In addition, we consider questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, we show that there are no natural non-trivial abelian subgroup of a group, nor a natural perfect quotient group of a group. More generally, we prove that natural subgroups, i.e. values of a sub-functor of the identity functor on groups, cannot all belong to a proper reflective subcategory. As an auxiliary result we prove that, for any non-trivial subfunctor <em>F</em> of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of <em>F</em>.</div><div>The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the ∞-category of spaces.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"675 ","pages":"Pages 273-288"},"PeriodicalIF":0.8,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825547","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":"Tilings of the sphere by congruent regular triangles and congruent rhombi","authors":"Qi Yuan,&nbsp;Erxiao Wang","doi":"10.1016/j.jalgebra.2025.03.033","DOIUrl":"10.1016/j.jalgebra.2025.03.033","url":null,"abstract":"<div><div>Abstract All edge-to-edge tilings of the sphere by congruent regular triangles and congruent rhombi are classified as: (1) a 1-parameter family of protosets each admitting a unique <span><math><mo>(</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>-tiling like a triangular prism; (2) a 1-parameter family of protosets each admitting 2 different <span><math><mo>(</mo><mn>8</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mn>6</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>-tilings like a cuboctahedron and a triangular orthobicupola respectively; (3) a sequence of protosets each admitting a unique <span><math><mo>(</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mo>(</mo><mn>6</mn><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>-tiling like a generalized anti-triangular prism for each <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>; (4) 26 sporadic protosets, among which nineteen admit a unique tiling, three admit 2 different tilings, one admits 3 different tilings, one admits 5 different tilings, two admit too many tilings to count (such polymorphic phenomena is unseen in monohedral spherical tilings with small total number of tiles). The moduli of parameterized tilings and all geometric data are provided.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 150-188"},"PeriodicalIF":0.8,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825479","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":"Finite non-parabolic subgroups of relatively hyperbolic groups","authors":"Oleg Bogopolski","doi":"10.1016/j.jalgebra.2025.03.036","DOIUrl":"10.1016/j.jalgebra.2025.03.036","url":null,"abstract":"<div><div>Let <em>G</em> be a group that is relatively hyperbolic with respect to a collection of subgroups <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>λ</mi><mo>∈</mo><mi>Λ</mi></mrow></msub></math></span>. Suppose that <em>G</em> is given by a finite relative presentation <span><math><mi>P</mi></math></span> with respect to this collection. We give an upper bound on the orders of finite non-parabolic subgroups of <em>G</em> in terms of some fundamental constants associated with <span><math><mi>P</mi></math></span>. This upper bound is computable if <em>G</em> is finitely generated and the word problem in each <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span>, <span><math><mi>λ</mi><mo>∈</mo><mi>Λ</mi></math></span>, is decidable.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 118-138"},"PeriodicalIF":0.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820624","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":"Non-surjectivity of the universal torsor evaluation map for homogeneous spaces","authors":"Mattia Pirani","doi":"10.1016/j.jalgebra.2025.03.032","DOIUrl":"10.1016/j.jalgebra.2025.03.032","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic zero, let <em>G</em> be a connected linear algebraic <em>K</em>-group, and let <em>H</em> be a connected closed subgroup of <em>G</em>. Let <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> be a smooth compactification of <span><math><mi>X</mi><mo>=</mo><mi>G</mi><mo>/</mo><mi>H</mi></math></span>, and let <span><math><mi>Y</mi><mover><mo>⟶</mo><mrow></mrow></mover><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> be the universal <em>S</em>-torsor with trivial fibre over the class of the identity of <em>G</em>. Colliot-Thélène and Kunyavskiĭ have shown that <em>S</em> is a flasque torus, and that the evaluation map <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>K</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>, induced by the universal torsor, is surjective when the field <em>K</em> is good; and the same is true when we restrict the evaluation map to the <em>K</em>-points of <em>X</em>. In this article, we establish that in cases where the field is not good, surjectivity may fail when the domain is <span><math><mi>X</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. We provide two concrete examples: one over a field of cohomological dimension 2 and the other over an arithmetic field, such as <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>(</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"675 ","pages":"Pages 237-272"},"PeriodicalIF":0.8,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815118","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":"Reflections to set-theoretic solutions of the Yang-Baxter equation","authors":"Andrea Albano,&nbsp;Marzia Mazzotta,&nbsp;Paola Stefanelli","doi":"10.1016/j.jalgebra.2025.03.034","DOIUrl":"10.1016/j.jalgebra.2025.03.034","url":null,"abstract":"<div><div>The main aim of this paper is to determine <em>reflections</em> to bijective and non-degenerate solutions of the Yang-Baxter equation, by exploring their connections with their derived solutions. This is motivated by a recent description of left non-degenerate solutions in terms of a family of automorphisms of their associated left rack. In some cases, we show that the study of reflections for bijective and non-degenerate solutions can be reduced to those of derived type. Moreover, we extend some results obtained in the literature for reflections of involutive non-degenerate solutions to more arbitrary solutions. Besides, we provide ways for defining reflections for solutions obtained by employing some classical construction techniques of solutions. Finally, we gather some numerical data on reflections for bijective non-degenerate solutions associated with skew braces of small order.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 106-138"},"PeriodicalIF":0.8,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808041","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":"q-bic forms","authors":"Raymond Cheng","doi":"10.1016/j.jalgebra.2025.03.031","DOIUrl":"10.1016/j.jalgebra.2025.03.031","url":null,"abstract":"<div><div>A <em>q-bic form</em> is a pairing <span><math><mi>V</mi><mo>×</mo><mi>V</mi><mo>→</mo><mi>k</mi></math></span> that is linear in the second variable and <em>q</em>-power Frobenius linear in the first; here, <em>V</em> is a vector space over a field <strong>k</strong> containing the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. This article develops a geometric theory of <em>q</em>-bic forms in the spirit of that of bilinear forms. I find two filtrations intrinsically attached to a <em>q</em>-bic form, with which I define a series of numerical invariants. These are used to classify, study automorphism group schemes of, and describe specialization relations in the parameter space of <em>q</em>-bic forms.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"675 ","pages":"Pages 196-236"},"PeriodicalIF":0.8,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807157","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":"Character of irreducible representations restricted to finite order elements - an asymptotic formula","authors":"Shrawan Kumar ,&nbsp;Dipendra Prasad","doi":"10.1016/j.jalgebra.2025.03.030","DOIUrl":"10.1016/j.jalgebra.2025.03.030","url":null,"abstract":"<div><div>Let <em>G</em> be a connected reductive group over the complex numbers and let <span><math><mi>T</mi><mo>⊂</mo><mi>G</mi></math></span> be a maximal torus. For any <span><math><mi>t</mi><mo>∈</mo><mi>T</mi></math></span> of finite order and any irreducible representation <span><math><mi>V</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> of <em>G</em> of highest weight <em>λ</em>, we determine the character <span><math><mi>ch</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>V</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>)</mo></math></span> by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety <span><math><msup><mrow><mo>(</mo><mi>G</mi><mo>/</mo><mi>P</mi><mo>)</mo></mrow><mrow><mi>t</mi></mrow></msup><mo>⊂</mo><mi>G</mi><mo>/</mo><mi>P</mi></math></span> (for any parabolic subgroup <em>P</em>). This together with Wirtinger's theorem gives an asymptotic formula for <span><math><mi>ch</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>V</mi><mo>(</mo><mi>n</mi><mi>λ</mi><mo>)</mo><mo>)</mo></math></span> when <em>n</em> goes to infinity.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 69-81"},"PeriodicalIF":0.8,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791645","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":"Lusztig's positive root vectors and a Dolbeault complex for the A-series full quantum flag manifolds","authors":"Réamonn Ó Buachalla,&nbsp;Petr Somberg","doi":"10.1016/j.jalgebra.2025.03.035","DOIUrl":"10.1016/j.jalgebra.2025.03.035","url":null,"abstract":"<div><div>For the Drinfeld–Jimbo quantum enveloping algebra <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>, we show that the span of Lusztig's positive root vectors, with respect to Littlemann's nice reduced decompositions of the longest element of the Weyl group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, form quantum tangent spaces for the full quantum flag manifold <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>. The associated differential calculi are direct <em>q</em>-deformations of the anti-holomorphic Dolbeault complex of the classical full flag manifold <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. As an application we establish a quantum Borel–Weil theorem for <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>, giving a noncommutative differential geometric realisation of all the finite-dimensional type-1 irreducible representations of <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>. Restricting this differential calculus to the quantum Grassmannians is shown to reproduce the celebrated Heckenberger–Kolb anti-holomorphic Dolbeault complex. Lusztig's positive root vectors for non-nice decompositions of the longest element of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> are examined for low orders, and are exhibited to either not give tangents spaces, or to produce differential calculi of non-classical dimension.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 1-73"},"PeriodicalIF":0.8,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807116","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}