Pub Date : 2024-01-17DOI: 10.1142/s0218196724500073
Carlos A. M. Andr'e, Ana L. Branco Correia, Joao Dias
Let $mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+mathcal{A}(q)$. On the other hand, let $Bbbk$ denote the algebraic closure of $mathbb{F}_{q}$, and let $mathcal{A} = mathcal{A}(q) otimes_{mathbb{F}_{q}} Bbbk$. Then $G = 1+mathcal{A}$ is an algebraic group over $Bbbk$ equipped with an $mathbb{F}_{q}$-rational structure given by the usual Frobenius map $F:Gto G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{F}$. For every $n in mathbb{N}$, the $n$th power $F^{n}:Gto G$ is also a Frobenius map, and $G^{F^{n}}$ identifies with $G(q^{n}) = 1 + mathcal{A}(q^{n})$. The Frobenius map restricts to a group automorphism $F:G(q^{n})to G(q^{n})$, and hence it acts on the set of irreducible characters of $G(q^{n})$. Shintani descent provides a method to compare $F$-invariant irreducible characters of $G(q^{n})$ and irreducible characters of $G(q)$. In this paper, we show that it also provides a uniform way of studying supercharacters of $G(q^{n})$ for $n in mathbb{N}$. These groups form an inductive system with respect to the inclusion maps $G(q^{m}) to G(q^{n})$ whenever $m mid n$, and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group $G$. Indeed, we show that Shintani descent permits the definition of a certain ``superdual algebra'' which encodes information about the supercharacters of $G(q^{n})$ for $n in mathbb{N}$.
{"title":"Shintani descent for standard supercharacters of algebra groups","authors":"Carlos A. M. Andr'e, Ana L. Branco Correia, Joao Dias","doi":"10.1142/s0218196724500073","DOIUrl":"https://doi.org/10.1142/s0218196724500073","url":null,"abstract":"Let $mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+mathcal{A}(q)$. On the other hand, let $Bbbk$ denote the algebraic closure of $mathbb{F}_{q}$, and let $mathcal{A} = mathcal{A}(q) otimes_{mathbb{F}_{q}} Bbbk$. Then $G = 1+mathcal{A}$ is an algebraic group over $Bbbk$ equipped with an $mathbb{F}_{q}$-rational structure given by the usual Frobenius map $F:Gto G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{F}$. For every $n in mathbb{N}$, the $n$th power $F^{n}:Gto G$ is also a Frobenius map, and $G^{F^{n}}$ identifies with $G(q^{n}) = 1 + mathcal{A}(q^{n})$. The Frobenius map restricts to a group automorphism $F:G(q^{n})to G(q^{n})$, and hence it acts on the set of irreducible characters of $G(q^{n})$. Shintani descent provides a method to compare $F$-invariant irreducible characters of $G(q^{n})$ and irreducible characters of $G(q)$. In this paper, we show that it also provides a uniform way of studying supercharacters of $G(q^{n})$ for $n in mathbb{N}$. These groups form an inductive system with respect to the inclusion maps $G(q^{m}) to G(q^{n})$ whenever $m mid n$, and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group $G$. Indeed, we show that Shintani descent permits the definition of a certain ``superdual algebra'' which encodes information about the supercharacters of $G(q^{n})$ for $n in mathbb{N}$.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140504684","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 : 2024-01-12DOI: 10.1142/s0218196724500036
Ilia Ponomarenko, A. Vasil’ev
{"title":"On computing the closures of solvable permutation groups","authors":"Ilia Ponomarenko, A. Vasil’ev","doi":"10.1142/s0218196724500036","DOIUrl":"https://doi.org/10.1142/s0218196724500036","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139532622","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 : 2024-01-12DOI: 10.1142/s0218196724500048
Edouard Feingesicht
{"title":"Dehornoy's class and sylows for set-theoretical solutions of the yang baxter equation","authors":"Edouard Feingesicht","doi":"10.1142/s0218196724500048","DOIUrl":"https://doi.org/10.1142/s0218196724500048","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139624103","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 : 2023-12-08DOI: 10.1142/s0218196723500698
M. Al-Tahan, B. Davvaz, P. Harikrishnan, P. Pallavi
{"title":"Subpolygroup commutativity degree of finite extension polygroup","authors":"M. Al-Tahan, B. Davvaz, P. Harikrishnan, P. Pallavi","doi":"10.1142/s0218196723500698","DOIUrl":"https://doi.org/10.1142/s0218196723500698","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139011475","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 : 2023-12-08DOI: 10.1142/s0218196723500686
H. A. Çoban
{"title":"Detecting similarities of rational plane curves using complex differential invariants","authors":"H. A. Çoban","doi":"10.1142/s0218196723500686","DOIUrl":"https://doi.org/10.1142/s0218196723500686","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139011530","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 : 2023-12-01DOI: 10.1142/s0218196723990011
{"title":"Author index Volume 33 (2023)","authors":"","doi":"10.1142/s0218196723990011","DOIUrl":"https://doi.org/10.1142/s0218196723990011","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139194284","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 : 2023-11-10DOI: 10.1142/s0218196723500674
Jesse Lansdown
Using the classification of transitive groups of degree $n$, for $2 leqslant n leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 leqslant n leqslant 48$. The results of this classification are made available as a supplementary database.
{"title":"A census of small Schurian association schemes","authors":"Jesse Lansdown","doi":"10.1142/s0218196723500674","DOIUrl":"https://doi.org/10.1142/s0218196723500674","url":null,"abstract":"Using the classification of transitive groups of degree $n$, for $2 leqslant n leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 leqslant n leqslant 48$. The results of this classification are made available as a supplementary database.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191733","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 : 2023-11-10DOI: 10.1142/s0218196723500662
C. E. Kofinas
{"title":"Symmetric polynomials in free centre-by-metabelian lie algebras of rank 2","authors":"C. E. Kofinas","doi":"10.1142/s0218196723500662","DOIUrl":"https://doi.org/10.1142/s0218196723500662","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191022","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 : 2023-11-09DOI: 10.1142/s0218196723500650
Claudemir Fideles, Alan Guimaraes
Let [Formula: see text] be the infinite-dimensional Grassmann algebra over an infinite field [Formula: see text] of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of [Formula: see text] equipped with its [Formula: see text] and [Formula: see text]-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.
设[公式:见文]为特征不等于2的无限域上的无限维Grassmann代数[公式:见文]。本文的主要目的是描述[公式:见文]的分级多项式恒等式的[理想]和[公式:见文]的中心多项式的[公式:见文]空间及其[公式:见文]和[公式:见文]诱导的[公式:见文]分级。因此，我们推广了[A]的结果。brand， C. Fidelis和A. guimar，[公式:见文本]-格拉斯曼代数的完全支持度评分，数学学报，31 (2022):332-353;C. Fidelis, A. guimar es和P. Koshlukov，关于[公式:见文本]的注解-关于Grassmann代数和初等数论的评分，线性多线性代数，71(7)(2023)1244-1264]。
{"title":"ℤ-Gradings on the Grassmann Algebra Over Infinite Fields: Graded Identities and Central Polynomials","authors":"Claudemir Fideles, Alan Guimaraes","doi":"10.1142/s0218196723500650","DOIUrl":"https://doi.org/10.1142/s0218196723500650","url":null,"abstract":"Let [Formula: see text] be the infinite-dimensional Grassmann algebra over an infinite field [Formula: see text] of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of [Formula: see text] equipped with its [Formula: see text] and [Formula: see text]-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191123","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 : 2023-09-22DOI: 10.1142/s0218196723500649
Alexander G. Melnikov, Keng Meng Ng
. We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arith-metical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every effectively locally compact tdlc group is topologically isomorphic to a computable tdlc group; all notions will be clarified. The result is perhaps unexpected since the hypothesis of the theorem may seem rather weak.
{"title":"Separating Notions in Effective Topology","authors":"Alexander G. Melnikov, Keng Meng Ng","doi":"10.1142/s0218196723500649","DOIUrl":"https://doi.org/10.1142/s0218196723500649","url":null,"abstract":". We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arith-metical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every effectively locally compact tdlc group is topologically isomorphic to a computable tdlc group; all notions will be clarified. The result is perhaps unexpected since the hypothesis of the theorem may seem rather weak.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099404","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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