Journal of Algebra最新文献

{"title":"Tied–boxed algebras","authors":"Diego Arcis ,&nbsp;Jorge Espinoza","doi":"10.1016/j.jalgebra.2025.07.039","DOIUrl":"10.1016/j.jalgebra.2025.07.039","url":null,"abstract":"<div><div>We introduce two new algebras that we call <em>tied–boxed Hecke algebra</em> and <em>tied–boxed Temperley–Lieb algebra</em>. The first one is a subalgebra of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the second one is a tied-version of the well known Temperley–Lieb algebra. We study their representation theory and give cellular bases for them. Furthermore, we explore a strong connection between the tied–boxed Temperley–Lieb algebra and the so-called partition Temperley–Lieb algebra given by Juyumaya. Also, we show that both structures inherit diagrammatic interpretations from a new class of monoids that we call <em>boxed ramified monoids</em>. Additionally, we give presentations for the singular part of the ramified symmetric monoid and for the boxed ramified monoid associated to the Brauer monoid.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 112-159"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831470","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":"Tensor extriangulated categories","authors":"Raphael Bennett-Tennenhaus ,&nbsp;Isambard Goodbody ,&nbsp;Janina C. Letz ,&nbsp;Amit Shah","doi":"10.1016/j.jalgebra.2025.07.041","DOIUrl":"10.1016/j.jalgebra.2025.07.041","url":null,"abstract":"<div><div>A tensor extriangulated category is an extriangulated category with a symmetric monoidal structure that is compatible with the extriangulated structure. To this end we define a notion of a biextriangulated functor <span><math><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>C</mi></math></span>, with compatibility conditions between the components. We have two versions of compatibility conditions, the stronger depending on the higher extensions of the extriangulated categories. We give many examples of tensor extriangulated categories. Finally, we generalise Balmer's classification of thick tensor ideals to tensor extriangulated categories.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 361-405"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831475","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":"Equational theories of the Boolean matrix monoid BRn with involutions","authors":"Wen-Ting Zhang,&nbsp;Meng Gao,&nbsp;Yan-Feng Luo","doi":"10.1016/j.jalgebra.2025.07.045","DOIUrl":"10.1016/j.jalgebra.2025.07.045","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the monoid of all &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; Boolean matrices with 1s on the main diagonal. It is known that &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; can be identified with the monoid of all reflexive binary relations on an &lt;em&gt;n&lt;/em&gt;-element set under composition. The monoid &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; admits two natural unary operations: the transposition &lt;sup&gt;&lt;em&gt;T&lt;/em&gt;&lt;/sup&gt; and the skew transposition &lt;sup&gt;&lt;em&gt;D&lt;/em&gt;&lt;/sup&gt;, which makes &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; an involutory monoid. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;) be the submonoid of &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; consisting of all &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; upper triangular Boolean matrices with 1s on the main diagonal (resp. all convex Boolean matrices) and &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; the monoid of all convex upper triangular Boolean matrices. Denote by &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; the double Catalan monoid which is a submonoid of &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;In this paper, we explore equational theories of the involutory monoids &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Firstly, we have completely solved the finite basis problems for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. It is shown that the involutory monoid &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is finitely based if and only if &lt;sp","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 225-270"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831472","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":"Characters of Ariki–Koike algebras","authors":"Deke Zhao","doi":"10.1016/j.jalgebra.2025.07.023","DOIUrl":"10.1016/j.jalgebra.2025.07.023","url":null,"abstract":"<div><div>In this paper, we prove the Regev formula for the characters of the Ariki–Koike algebras by applying the Schur–Sergeev reciprocity between quantum superalgebras and Ariki–Koike algebras. As a corollary, we derive the Regev formula for the characters of the complex reflection group of type <span><math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, generalizing the Regev formula for the symmetric groups due to A. Regev (2013) <span><span>[19]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 773-791"},"PeriodicalIF":0.8,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144813982","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":"Brauer's 14th Problem and Dyson's tenfold way","authors":"Dmitriy Rumynin ,&nbsp;James Taylor","doi":"10.1016/j.jalgebra.2025.07.026","DOIUrl":"10.1016/j.jalgebra.2025.07.026","url":null,"abstract":"<div><div>We consider Brauer's 14th Problem in the context of <em>Real</em> structures on finite groups and their antilinear representations. The problem is to count the number of characters of each different type using “group theory”. While Brauer's original problem deals only with three types (real, complex and quaternionic), here we consider the ten types coming from Dyson's tenfold way.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 463-473"},"PeriodicalIF":0.8,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144861234","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":"On the fibers of the principal minor map and an application to stable polynomials","authors":"Abeer Al Ahmadieh","doi":"10.1016/j.jalgebra.2025.07.030","DOIUrl":"10.1016/j.jalgebra.2025.07.030","url":null,"abstract":"<div><div>This paper explores the fibers of the principal minor map over an arbitrary field. The principal minor map assigns to each <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix the <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>-vector of its principal minors. In 1984, Hartfiel and Loewy proposed a condition that was sufficient to ensure that the fiber of the principal minor map is a single point up to diagonal equivalence. Loewy later improved upon this condition in 1986. In this paper, we provide a necessary and sufficient condition for the fiber to be a point up to diagonal equivalence. Additionally, we establish a connection between the reducibility of a matrix and the reducibility of its determinantal representation. Using this connection, we fully characterize the fibers that contain a symmetric or Hermitian matrix in the space of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over any field. We also use these techniques to answer a question of Borcea, Brändén, and Liggett concerning real stable matrices.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 46-61"},"PeriodicalIF":0.8,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831467","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":"Strong u-invariant and period-index bound for complete ultrametric fields","authors":"Shilpi Mandal","doi":"10.1016/j.jalgebra.2025.07.021","DOIUrl":"10.1016/j.jalgebra.2025.07.021","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;k&lt;/em&gt; be a complete ultrametric field with &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;dim&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; finite, with residue field &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;, and char&lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We prove that &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the strong &lt;em&gt;u&lt;/em&gt;-invariant of &lt;em&gt;k&lt;/em&gt;, then we further show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;em&gt;l&lt;/em&gt; be a prime such that the char&lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. If the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;Br&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtext&gt;dim&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, then we also show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;Br&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtext&gt;dim&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;em&gt;C&lt;/em&gt; be a curve over &lt;em&gt;k&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Then we show that any quadratic form over &lt;em&gt;F&lt;/em&gt; with dimension &lt;span&gt;&lt;math&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is isotropic over &lt;em&gt;F&lt;/em&gt;. We further show that if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;Br&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtext&gt;dim&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;Br&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtext&gt;dim&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;Br&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtex","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 447-462"},"PeriodicalIF":0.8,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144861233","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 Hesse pencil of plane curves and osculating conics","authors":"Ewelina Nawara","doi":"10.1016/j.jalgebra.2025.07.040","DOIUrl":"10.1016/j.jalgebra.2025.07.040","url":null,"abstract":"<div><div>In this paper, we revisit the classical problem of determining osculating conics and sextactic points for a given algebraic curve. Our focus is on a particular family of plane cubic curves known as the Hesse pencil. By employing classical tools from projective differential geometry, we derive explicit coordinates for these special points. The resulting formulas not only clarify previous approaches but also lead to the construction of new families of free and nearly free curves, extending recent findings the freeness of curves.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 536-549"},"PeriodicalIF":0.8,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865367","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":"Multiplicative character sums and Kloosterman sheaves","authors":"Nicholas M. Katz ,&nbsp;Pham Huu Tiep","doi":"10.1016/j.jalgebra.2025.07.019","DOIUrl":"10.1016/j.jalgebra.2025.07.019","url":null,"abstract":"<div><div>We are given a prime <em>p</em>, a power <em>q</em> of <em>p</em>, and a prime to <em>p</em> integer <em>a</em> with <span><math><mi>q</mi><mo>&gt;</mo><mi>a</mi><mo>≥</mo><mn>2</mn></math></span>. For a nontrivial multiplicative character <em>χ</em>, we consider the one parameter family of character sums<span><span><span><math><mi>t</mi><mo>↦</mo><mo>−</mo><munder><mo>∑</mo><mrow><mi>x</mi></mrow></munder><mi>χ</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>a</mi></mrow></msup><mo>−</mo><mi>t</mi><mo>)</mo><mo>,</mo></math></span></span></span> which are the traces of a local system on the <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>χ</mi><mo>)</mo></math></span> of nonzero <em>t</em>'s. We show that this local system is the pullback of a Kloosterman sheaf <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>ρ</mi></mrow></msub></math></span> (any <em>ρ</em> with <span><math><msup><mrow><mi>ρ</mi></mrow><mrow><mi>q</mi><mo>−</mo><mi>a</mi></mrow></msup><mo>=</mo><mi>χ</mi></math></span>), and determine the geometric monodromy group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>geom</mi></mrow></msub></math></span> of this <span><math><mi>K</mi></math></span>. We also determine <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>geom</mi></mrow></msub></math></span> for the universal family <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>χ</mi><mo>,</mo><mi>e</mi></mrow></msub></math></span> of sums <span><math><mo>−</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>χ</mi><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span>, as <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>e</mi></mrow></msub></math></span> runs over degree <em>e</em> polynomials with all distinct roots. These local systems <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>χ</mi><mo>,</mo><mi>e</mi></mrow></msub></math></span> were the main focus of <span><span>[14, Chapter 4]</span></span>, and our new results for <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>χ</mi><mo>,</mo><mi>e</mi></mrow></msub></math></span> are the complete determination of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>geom</mi></mrow></msub></math></span> in the cases where <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>geom</mi></mrow></msub></math></span> is finite.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 738-772"},"PeriodicalIF":0.8,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810643","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":"Positivity preservers over finite fields","authors":"Dominique Guillot ,&nbsp;Himanshu Gupta ,&nbsp;Prateek Kumar Vishwakarma ,&nbsp;Chi Hoi Yip","doi":"10.1016/j.jalgebra.2025.07.016","DOIUrl":"10.1016/j.jalgebra.2025.07.016","url":null,"abstract":"<div><div>We resolve an algebraic version of Schoenberg's celebrated theorem [<em>Duke Math. J.</em>, 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider matrices with entries in a finite field and obtain a complete characterization of such preservers for matrices of a fixed dimension. When the dimension of the matrices is at least 3, we prove that, surprisingly, the positivity preservers are precisely the positive multiples of the field's automorphisms. We also obtain characterizations of preservers in the significantly more challenging dimension 2 case over a finite field with <em>q</em> elements, unless <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span> and <em>q</em> is not a square. Our proofs build on several novel connections between positivity preservers and field automorphisms via the works of Weil, Carlitz, and Muzychuk-Kovács, and via the structure of cliques in Paley graphs.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 479-523"},"PeriodicalIF":0.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144767000","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}