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On the cardinality of irredundant and minimal bases of finite permutation groups 论有限置换群的非冗余基和最小基的心性
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-07-05 DOI: 10.1007/s10801-024-01343-w
Francesca Dalla Volta, Fabio Mastrogiacomo, Pablo Spiga
{"title":"On the cardinality of irredundant and minimal bases of finite permutation groups","authors":"Francesca Dalla Volta, Fabio Mastrogiacomo, Pablo Spiga","doi":"10.1007/s10801-024-01343-w","DOIUrl":"https://doi.org/10.1007/s10801-024-01343-w","url":null,"abstract":"<p>Given a finite permutation group <i>G</i> with domain <span>(Omega )</span>, we associate two subsets of natural numbers to <i>G</i>, namely <span>({mathcal {I}}(G,Omega ))</span> and <span>({mathcal {M}}(G,Omega ))</span>, which are the sets of cardinalities of all the irredundant and minimal bases of <i>G</i>, respectively. We prove that <span>({mathcal {I}}(G,Omega ))</span> is an interval of natural numbers, whereas <span>({mathcal {M}}(G,Omega ))</span> may not necessarily form an interval. Moreover, for a given subset of natural numbers <span>(X subseteq {mathbb {N}})</span>, we provide some conditions on <i>X</i> that ensure the existence of both intransitive and transitive groups <i>G</i> such that <span>({mathcal {I}}(G,Omega ) = X)</span> and <span>({mathcal {M}}(G,Omega ) = X)</span>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"35 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550302","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}
引用次数: 0
Transitive generalized toggle groups containing a cycle 包含一个循环的传递广义切换群
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-07-05 DOI: 10.1007/s10801-024-01348-5
Jonathan S. Bloom, Dan Saracino
{"title":"Transitive generalized toggle groups containing a cycle","authors":"Jonathan S. Bloom, Dan Saracino","doi":"10.1007/s10801-024-01348-5","DOIUrl":"https://doi.org/10.1007/s10801-024-01348-5","url":null,"abstract":"<p>In (Striker in Discret Math Theor Comput Sci 20, 2018), Striker generalized Cameron and Fon-Der-Flaass’s notion of a toggle group. In this paper, we begin the study of transitive generalized toggle groups that contain a cycle. We first show that if such a group has degree <i>n</i> and contains a transposition or a 3-cycle, then the group contains <span>(A_n)</span>. Using the result about transpositions, we then prove that a transitive generalized toggle group that contains a short cycle must be primitive. Employing a result of Jones (Bull Aust Math Soc 89(1):159-165, 2014), which relies on the classification of the finite simple groups, we conclude that any transitive generalized toggle group of degree <i>n</i> that contains a cycle with at least 3 fixed points must also contain <span>(A_n)</span>. Finally, we look at imprimitive generalized toggle groups containing a long cycle and show that they decompose into a direct product of primitive generalized toggle groups each containing a long cycle.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567650","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}
引用次数: 0
The maximum number of homogeneous weights of linear codes over chain rings 链环上线性编码的最大同权数
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-07-02 DOI: 10.1007/s10801-024-01347-6
Minjia Shi, Tingting Tong, Thomas Honold, Patrick Solé
{"title":"The maximum number of homogeneous weights of linear codes over chain rings","authors":"Minjia Shi, Tingting Tong, Thomas Honold, Patrick Solé","doi":"10.1007/s10801-024-01347-6","DOIUrl":"https://doi.org/10.1007/s10801-024-01347-6","url":null,"abstract":"<p>The problem of determining the largest possible number of distinct Hamming weights in several classes of codes over finite fields was studied recently in several papers (Shi et al. in Des Codes Cryptogr 87(1):87–95, 2019, in IEEE Trans Inf Theory 66(11):6855–6862, 2020; Chen et al. in IEEE Trans Inf Theory 69(2):995–1004, 2022). A further problem is to find the minimum length of codes meeting those bounds with equality. These two questions are extended here to linear codes over chain rings for the homogeneous weight. An explicit upper bound is given for codes of given type and arbitrary length as a function of the residue field size. This bound is then shown to be tight by an argument based on Hjemslev geometries. The second question is studied for chain rings with residue field of order two.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"65 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513883","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}
引用次数: 0
The sequentially Cohen–Macaulay property of edge ideals of edge-weighted graphs 边缘加权图的边缘理想的顺序科恩-麦考莱特性
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-07-02 DOI: 10.1007/s10801-024-01344-9
Ly Thi Kieu Diem, Nguyên Công Minh, Thanh Vu
{"title":"The sequentially Cohen–Macaulay property of edge ideals of edge-weighted graphs","authors":"Ly Thi Kieu Diem, Nguyên Công Minh, Thanh Vu","doi":"10.1007/s10801-024-01344-9","DOIUrl":"https://doi.org/10.1007/s10801-024-01344-9","url":null,"abstract":"<p>Let <span>(I(G,textbf{w}))</span> be the edge ideal of an edge-weighted graph <span>((G,textbf{w}))</span>. We prove that <span>(I(G,textbf{w}))</span> is sequentially Cohen–Macaulay for all weight functions <span>(textbf{w})</span> if and only if <i>G</i> is a Woodroofe graph.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"10 2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513884","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}
引用次数: 0
Generalization of the $$epsilon $$ -BBS and the Schensted insertion algorithm $$epsilon$$-BBS和申斯泰德插入算法的一般化
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-07-01 DOI: 10.1007/s10801-024-01338-7
Katsuki Kobayashi, Satoshi Tsujimoto
{"title":"Generalization of the $$epsilon $$ -BBS and the Schensted insertion algorithm","authors":"Katsuki Kobayashi, Satoshi Tsujimoto","doi":"10.1007/s10801-024-01338-7","DOIUrl":"https://doi.org/10.1007/s10801-024-01338-7","url":null,"abstract":"<p>The <span>(epsilon )</span>-BBS is the family of solitonic cellular automata obtained via the ultradiscretization of the elementary Toda orbits, which is a parametrized family of integrable systems unifying the Toda equation and the relativistic Toda equation. In this paper, we derive the <span>(epsilon )</span>-BBS with many kinds of balls and give its conserved quantities by the Schensted insertion algorithm which is introduced in combinatorics. To prove this, we extend birational transformations of the continuous elementary Toda orbits to the discrete hungry elementary Toda orbits.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"20 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529706","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}
引用次数: 0
Cohomology-developed matrices: constructing families of weighing matrices and automorphism actions 同调发展矩阵：构建称重矩阵族和自动态作用
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-06-27 DOI: 10.1007/s10801-024-01346-7
Assaf Goldberger, Giora Dula
{"title":"Cohomology-developed matrices: constructing families of weighing matrices and automorphism actions","authors":"Assaf Goldberger, Giora Dula","doi":"10.1007/s10801-024-01346-7","DOIUrl":"https://doi.org/10.1007/s10801-024-01346-7","url":null,"abstract":"<p>The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley conference, the projective space, the Grassmannian, and the flag variety weighing matrices. We develop a general theory relying on low-dimensional group cohomology for constructing automorphism group actions and in turn obtain structured matrices that we call <i>cohomology-developed matrices</i>. This ‘cohomology development’ generalizes the cocyclic and group developments. The algebraic structure of modules of cohomology-developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of <i>quasiproducts</i>, which is a generalization of the Kronecker product.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"11 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508705","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}
引用次数: 0
A local approach to stability groups 稳定小组的地方方法
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-06-25 DOI: 10.1007/s10801-024-01345-8
Martin L. Newell, Marco Trombetti
{"title":"A local approach to stability groups","authors":"Martin L. Newell, Marco Trombetti","doi":"10.1007/s10801-024-01345-8","DOIUrl":"https://doi.org/10.1007/s10801-024-01345-8","url":null,"abstract":"<p>In this short note we prove a local version of Philip Hall’s theorem on the nilpotency of the stability group of a chain of subgroups by only using elementary commutator calculus (Hall’s theorem is a direct consequence of our result). This provides a new way of dealing with stability groups.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"115 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508702","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}
引用次数: 0
Minimal degrees for faithful permutation representations of groups of order $$p^6$$ where p is an odd prime p^6$$阶（p为奇数素数）群的忠实置换表示的最小度数
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-06-22 DOI: 10.1007/s10801-024-01331-0
E. A. O’Brien, Sunil Kumar Prajapati, Ayush Udeep
{"title":"Minimal degrees for faithful permutation representations of groups of order $$p^6$$ where p is an odd prime","authors":"E. A. O’Brien, Sunil Kumar Prajapati, Ayush Udeep","doi":"10.1007/s10801-024-01331-0","DOIUrl":"https://doi.org/10.1007/s10801-024-01331-0","url":null,"abstract":"<p>We determine the minimal degree of a faithful permutation representation for each group of order <span>(p^6)</span> where <i>p</i> is an odd prime. We also record how to obtain such a representation.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The number of spanning trees in $$K_n$$ -complement of a bipartite graph 双方形图的$K_n$$补集中生成树的数目
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-06-01 DOI: 10.1007/s10801-024-01341-y
Helin Gong, Yu Gong, Jun Ge
{"title":"The number of spanning trees in $$K_n$$ -complement of a bipartite graph","authors":"Helin Gong, Yu Gong, Jun Ge","doi":"10.1007/s10801-024-01341-y","DOIUrl":"https://doi.org/10.1007/s10801-024-01341-y","url":null,"abstract":"<p>For a subgraph <i>G</i> of a complete graph <span>(K_n)</span>, the <span>(K_n)</span>-complement of <i>G</i>, denoted by <span>(K_n-G)</span>, is the graph obtained from <span>(K_n-G)</span> by removing all the edges of <i>G</i>. In this paper, we express the number of spanning trees of the <span>(K_n)</span>-complement <span>(K_n-G)</span> of a bipartite graph <i>G</i> in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of <i>G</i>, which are nonsingular, and we derive formulas of the number of spanning trees of <span>(K_n-G)</span> for various important classes of bipartite graphs <i>G</i>, some of which generalize some previous results.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195355","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}
引用次数: 0
The Ariki–Koike algebras and Rogers–Ramanujan type partitions 有熊小池代数和罗杰斯-拉马努扬类型分区
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-31 DOI: 10.1007/s10801-024-01340-z
Shane Chern, Zhitai Li, Dennis Stanton, Ting Xue, Ae Ja Yee
{"title":"The Ariki–Koike algebras and Rogers–Ramanujan type partitions","authors":"Shane Chern, Zhitai Li, Dennis Stanton, Ting Xue, Ae Ja Yee","doi":"10.1007/s10801-024-01340-z","DOIUrl":"https://doi.org/10.1007/s10801-024-01340-z","url":null,"abstract":"<p>Ariki and Mathas (Math Z 233(3):601–623, 2000) showed that the simple modules of the Ariki–Koike algebras <span>(mathcal {H}_{mathbb {C},v;Q_1,ldots , Q_m}big (G(m, 1, n)big ))</span> (when the parameters are roots of unity and <span>(v ne 1)</span>) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit their generating function relation for the <span>(v=-1)</span> case. In particular, this <span>(v=-1)</span> scenario is of special interest as the corresponding Kleshchev multipartitions are closely tied with generalized Rogers–Ramanujan type partitions when <span>(Q_1=cdots =Q_a=-1)</span> and <span>(Q_{a+1}=cdots =Q_m =1)</span>. Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for the above choice of parameters. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block, which are, as widely known, labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities for the <span>(m=2)</span> scenario.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"53 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195403","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}
引用次数: 0
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