Pub Date : 2024-05-07DOI: 10.1007/s10801-024-01319-w
Allen Herman
The Terwilliger algebras of asymmetric association schemes of rank 3, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension (4k+9) for some positive integer k. It is determined that asymmetric rank 3 association schemes of order up to 23 are determined up to combinatorial isomorphism by the list of their complex Terwilliger algebras at each vertex, but this is no longer true at order 27. To distinguish order 27 asymmetric rank 3 association schemes, it is shown using computer calculations that the list of rational Terwilliger algebras at each vertex will suffice.
{"title":"The Terwilliger algebras of doubly regular tournaments","authors":"Allen Herman","doi":"10.1007/s10801-024-01319-w","DOIUrl":"https://doi.org/10.1007/s10801-024-01319-w","url":null,"abstract":"<p>The Terwilliger algebras of asymmetric association schemes of rank 3, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension <span>(4k+9)</span> for some positive integer <i>k</i>. It is determined that asymmetric rank 3 association schemes of order up to 23 are determined up to combinatorial isomorphism by the list of their complex Terwilliger algebras at each vertex, but this is no longer true at order 27. To distinguish order 27 asymmetric rank 3 association schemes, it is shown using computer calculations that the list of rational Terwilliger algebras at each vertex will suffice.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-07DOI: 10.1007/s10801-024-01323-0
C. Dalfó, M. A. Fiol
The k-token graph (F_k(G)) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of (F_k(G)) equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of (F_k(G)) equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.
图 G 的 k 标记图 (F_k(G))是指其顶点是来自 G 的顶点的 k 子集的图,只要它们的对称差是 G 中的一对相邻顶点,其中的两个顶点就是相邻的。有人证明了 (F_k(G))的代数连通性等于 G 的代数连通性,证明中使用了加权图上的随机行走和交换过程。然而,目前还没有代数或组合证明,这将是该领域的一个重大突破。在本文中,我们用代数方法证明了对于新的无限图族,如树、一些有悬挂树的图和最小度足够大的图,(F_k(G))的代数连通性等于 G 的代数连通性。这些族的一些例子如下:鸡尾酒会图、循环补图和完整多方图。
{"title":"On the algebraic connectivity of some token graphs","authors":"C. Dalfó, M. A. Fiol","doi":"10.1007/s10801-024-01323-0","DOIUrl":"https://doi.org/10.1007/s10801-024-01323-0","url":null,"abstract":"<p>The <i>k</i>-token graph <span>(F_k(G))</span> of a graph <i>G</i> is the graph whose vertices are the <i>k</i>-subsets of vertices from <i>G</i>, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in <i>G</i>. It was proved that the algebraic connectivity of <span>(F_k(G))</span> equals the algebraic connectivity of <i>G</i> with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of <span>(F_k(G))</span> equals the one of <i>G</i> for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-06DOI: 10.1007/s10801-024-01324-z
Cai Heng Li, Yan Zhou Zhu
We introduce the concept of pseudocover, which is a counterpart of cover, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger–Xu’s graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by 32 has connected pseudocovers.
{"title":"Covers and pseudocovers of symmetric graphs","authors":"Cai Heng Li, Yan Zhou Zhu","doi":"10.1007/s10801-024-01324-z","DOIUrl":"https://doi.org/10.1007/s10801-024-01324-z","url":null,"abstract":"<p>We introduce the concept of <i>pseudocover</i>, which is a counterpart of <i>cover</i>, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger–Xu’s graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by 32 has connected pseudocovers.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-06DOI: 10.1007/s10801-024-01325-y
Saeid Azam
The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite-order Cartan automorphism is diagonal, and its corresponding combinatorial map is a character.
{"title":"Characters for extended affine Lie algebras: a combinatorial approach","authors":"Saeid Azam","doi":"10.1007/s10801-024-01325-y","DOIUrl":"https://doi.org/10.1007/s10801-024-01325-y","url":null,"abstract":"<p>The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite-order Cartan automorphism is diagonal, and its corresponding combinatorial map is a character.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"147 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-21DOI: 10.1007/s10801-024-01309-y
Dong-Qi Wan, Jianbing Liu, Jin Ho Kwak, Jin-Xin Zhou
Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by Kwak and Lee. For regular graph coverings, Kwak, Chun and Lee enumerated the isomorphism classes of graph coverings when the covering transformation group is a finite abelian or a dihedral group in Kwak et al. (SIAM J Discrete Math 11:273–285, 1998). In 2018, the isomorphism classes of graph coverings are enumerated when the covering transformation groups are (mathbb {Z}_2)-extensions of a cyclic group. As a continuation of this work, we enumerate the isomorphism classes of coverings of a graph when the covering transformation groups are (mathbb {Z}_p)-extensions of a cyclic group for an odd prime integer p.
枚举几类图覆盖的同构或等价类是枚举拓扑图理论的核心研究课题之一。1988 年,霍夫迈斯特(Hofmeister)枚举了图的双重覆盖,郭(Kwak)和李(Lee)将这项工作扩展到图的 n 重覆盖。对于规则图覆盖,Kwak、Chun 和 Lee 在 Kwak et al. (SIAM J Discrete Math 11:273-285, 1998) 中列举了当覆盖变换群是有限无边群或二面群时图覆盖的同构类。2018年,当覆盖变换群是一个循环群的(mathbb {Z}_2)-扩展时,图覆盖的同构类被列举出来。作为这项工作的延续,我们列举了当覆盖变换群是奇素数整数p的循环群的(mathbb {Z}_p)-扩展时,图覆盖的同构类。
{"title":"Enumerating regular graph coverings whose covering transformation groups are $$mathbb {Z}_p$$ -extensions of a cyclic group","authors":"Dong-Qi Wan, Jianbing Liu, Jin Ho Kwak, Jin-Xin Zhou","doi":"10.1007/s10801-024-01309-y","DOIUrl":"https://doi.org/10.1007/s10801-024-01309-y","url":null,"abstract":"<p>Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to <i>n</i>-fold coverings of a graph by Kwak and Lee. For <i>regular</i> graph coverings, Kwak, Chun and Lee enumerated the isomorphism classes of graph coverings when the covering transformation group is a finite abelian or a dihedral group in Kwak et al. (SIAM J Discrete Math 11:273–285, 1998). In 2018, the isomorphism classes of graph coverings are enumerated when the covering transformation groups are <span>(mathbb {Z}_2)</span>-extensions of a cyclic group. As a continuation of this work, we enumerate the isomorphism classes of coverings of a graph when the covering transformation groups are <span>(mathbb {Z}_p)</span>-extensions of a cyclic group for an odd prime integer <i>p</i>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140635448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-17DOI: 10.1007/s10801-024-01320-3
Lara Bossinger, Martina Lanini
We study the effect of Feigin’s flat degeneration of the type (text {A}) flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some degenerate Schubert varieties (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin’s Gröbner degeneration) with Richardson varieties in higher rank partial flag varieties.
{"title":"Following Schubert varieties under Feigin’s degeneration of the flag variety","authors":"Lara Bossinger, Martina Lanini","doi":"10.1007/s10801-024-01320-3","DOIUrl":"https://doi.org/10.1007/s10801-024-01320-3","url":null,"abstract":"<p>We study the effect of Feigin’s flat degeneration of the type <span>(text {A})</span> flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some <i>degenerate Schubert varieties</i> (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin’s Gröbner degeneration) with Richardson varieties in higher rank partial flag varieties.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"113 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-17DOI: 10.1007/s10801-024-01314-1
Sriparna Chattopadhyay, Kamal Lochan Patra, Binod Kumar Sahoo
Let R be a finite commutative ring with identity. We study the structure of the zero-divisor graph of R and then determine its vertex connectivity when: (i) R is a local principal ideal ring, and (ii) R is a finite direct product of local principal ideal rings. For such rings R, we also characterize the vertices of minimum degree and the minimum cut-sets of the zero-divisor graph of R.
设 R 是具有同一性的有限交换环。我们将研究 R 的零分维图的结构,然后确定其顶点连通性,前提是:(i) R 是局部主理想环;(ii) R 是局部主理想环的有限直积:(i) R 是局部主理想环,以及 (ii) R 是局部主理想环的有限直积。对于这样的环 R,我们还确定了 R 的零因子图的最小度顶点和最小切集的特征。
{"title":"On vertex connectivity of zero-divisor graphs of finite commutative rings","authors":"Sriparna Chattopadhyay, Kamal Lochan Patra, Binod Kumar Sahoo","doi":"10.1007/s10801-024-01314-1","DOIUrl":"https://doi.org/10.1007/s10801-024-01314-1","url":null,"abstract":"<p>Let <i>R</i> be a finite commutative ring with identity. We study the structure of the zero-divisor graph of <i>R</i> and then determine its vertex connectivity when: (i) <i>R</i> is a local principal ideal ring, and (ii) <i>R</i> is a finite direct product of local principal ideal rings. For such rings <i>R</i>, we also characterize the vertices of minimum degree and the minimum cut-sets of the zero-divisor graph of <i>R</i>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"89 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140627928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-09DOI: 10.1007/s10801-024-01317-y
Giovanni Longobardi, Corrado Zanella
In this paper, we present results concerning the stabilizer (G_f) in ({{,mathrm{{GL}},}}(2,q^n)) of the subspace (U_f={(x,f(x)):xin mathbb {F}_{q^n}}), f(x) a scattered linearized polynomial in (mathbb {F}_{q^n}[x]). Each (G_f) contains the (q-1) maps ((x,y)mapsto (ax,ay)), (ain mathbb {F}_{q}^*). By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in (G_f) are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that (|G_f|>q-1) have a standard form of type (sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}) for some s and t such that ((s,t)=1), (t>1) a divisor of n; (ii) this standard form is essentially unique; (iii) for (n>2) and (q>3), the translation plane (mathcal {A}_f) associated with f(x) admits nontrivial affine homologies if and only if (|G_f|>q-1), and in that case those with axis through the origin form two groups of cardinality ((q^t-1)/(q-1)) that exchange axes and coaxes; (iv) no plane of type (mathcal {A}_f), f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.
在本文中,我们提出了关于子空间 (U_f={(x,f(x)):x in mathbb {F}_{q^n}}), f(x) a scattered linearized polynomial in (mathbb {F}_{q^n}[x]).每个 G_f 都包含(q-1)映射((x,y)映射到(ax,ay)),(a 在 mathbb {F}_{q}^*) 中)。根据 Beard (Duke Math J, 39:313-321, 1972) 和 Willett (Duke Math J 40(3):701-704, 1973) 的结果,(G_f) 中的矩阵是同时可对角的。这有几个后果:(i) (|G_f|>q-1/)中的多项式对于某些 s 和 t 具有标准的 (sum_{j=0}^{n/t-1}a_jx^{q^{s+jt}}/)类型,即 ((s,t)=1/),(t>1/)是 n 的除数;(iii) 对于 (n>2) 和 (q>3), 与 f(x) 相关联的平移平面 (mathcal {A}_f) 允许非对称仿射同调,当且仅当 (|G_f|>;q-1),在这种情况下,那些轴通过原点的平面会形成两个交换轴和同轴的心数为((q^t-1)/(q-1))的群;(iv) 没有一个 f(x) 散点多项式不属于伪多径类型的 (mathcal {A}_f) 型平面是广义的安德烈平面。
{"title":"A standard form for scattered linearized polynomials and properties of the related translation planes","authors":"Giovanni Longobardi, Corrado Zanella","doi":"10.1007/s10801-024-01317-y","DOIUrl":"https://doi.org/10.1007/s10801-024-01317-y","url":null,"abstract":"<p>In this paper, we present results concerning the stabilizer <span>(G_f)</span> in <span>({{,mathrm{{GL}},}}(2,q^n))</span> of the subspace <span>(U_f={(x,f(x)):xin mathbb {F}_{q^n}})</span>, <i>f</i>(<i>x</i>) a scattered linearized polynomial in <span>(mathbb {F}_{q^n}[x])</span>. Each <span>(G_f)</span> contains the <span>(q-1)</span> maps <span>((x,y)mapsto (ax,ay))</span>, <span>(ain mathbb {F}_{q}^*)</span>. By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in <span>(G_f)</span> are simultaneously diagonalizable. This has several consequences: (<i>i</i>) the polynomials such that <span>(|G_f|>q-1)</span> have a standard form of type <span>(sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}})</span> for some <i>s</i> and <i>t</i> such that <span>((s,t)=1)</span>, <span>(t>1)</span> a divisor of <i>n</i>; (<i>ii</i>) this standard form is essentially unique; (<i>iii</i>) for <span>(n>2)</span> and <span>(q>3)</span>, the translation plane <span>(mathcal {A}_f)</span> associated with <i>f</i>(<i>x</i>) admits nontrivial affine homologies if and only if <span>(|G_f|>q-1)</span>, and in that case those with axis through the origin form two groups of cardinality <span>((q^t-1)/(q-1))</span> that exchange axes and coaxes; (<i>iv</i>) no plane of type <span>(mathcal {A}_f)</span>, <i>f</i>(<i>x</i>) a scattered polynomial not of pseudoregulus type, is a generalized André plane.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in ({1,2,dots , n-1}). Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.
{"title":"An ultimately periodic chain in the integral Lie ring of partitions","authors":"Riccardo Aragona, Roberto Civino, Norberto Gavioli","doi":"10.1007/s10801-024-01318-x","DOIUrl":"https://doi.org/10.1007/s10801-024-01318-x","url":null,"abstract":"<p>Given an integer <i>n</i>, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in <span>({1,2,dots , n-1})</span>. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"2013 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-06DOI: 10.1007/s10801-024-01315-0
Abstract
Let ({mathcal {D}}) be a non-trivial G-block-transitive 3-(v, k, 1) design, where (Tle G le textrm{Aut}(T)) for some finite non-abelian simple group T. It is proved that if T is a simple exceptional group of Lie type, then T is either the Suzuki group ({}^2B_2(q)) or (G_2(q)). Furthermore, if (T={}^2B_2(q)) then the design ({mathcal {D}}) has parameters (v=q^2+1) and (k=q+1), and so ({mathcal {D}}) is an inverse plane of order q, and if (T=G_2(q)) then the point stabilizer in T is either (textrm{SL}_3(q).2) or (textrm{SU}_3(q).2), and the parameter k satisfies very restricted conditions.
摘要 让 ({mathcal {D}}) 是一个非难的 G 块传递的 3-(v,k,1)设计,其中 (Tle G le textrm{Aut}(T)) 对于某个有限的非阿贝尔简单群 T。此外,如果 (T={}^2B_2(q)) 那么设计 ({mathcal {D}}) 有参数 (v=q^2+1) 和 (k=q+1) ,所以 ({mathcal {D}}) 是一个 q 阶的反平面,如果 (T=G_2(q)) 那么 T 中的点稳定器要么是 (textrm{SL}_3(q).2) 或者 (textrm{SU}_3(q).参数 k 满足非常有限的条件。
{"title":"Block-transitive 3-(v, k, 1) designs on exceptional groups of Lie type","authors":"","doi":"10.1007/s10801-024-01315-0","DOIUrl":"https://doi.org/10.1007/s10801-024-01315-0","url":null,"abstract":"<h3>Abstract</h3> <p>Let <span> <span>({mathcal {D}})</span> </span> be a non-trivial <em>G</em>-block-transitive 3-(<em>v</em>, <em>k</em>, 1) design, where <span> <span>(Tle G le textrm{Aut}(T))</span> </span> for some finite non-abelian simple group <em>T</em>. It is proved that if <em>T</em> is a simple exceptional group of Lie type, then <em>T</em> is either the Suzuki group <span> <span>({}^2B_2(q))</span> </span> or <span> <span>(G_2(q))</span> </span>. Furthermore, if <span> <span>(T={}^2B_2(q))</span> </span> then the design <span> <span>({mathcal {D}})</span> </span> has parameters <span> <span>(v=q^2+1)</span> </span> and <span> <span>(k=q+1)</span> </span>, and so <span> <span>({mathcal {D}})</span> </span> is an inverse plane of order <em>q</em>, and if <span> <span>(T=G_2(q))</span> </span> then the point stabilizer in <em>T</em> is either <span> <span>(textrm{SL}_3(q).2)</span> </span> or <span> <span>(textrm{SU}_3(q).2)</span> </span>, and the parameter <em>k</em> satisfies very restricted conditions.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"56 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}