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Alternating groups as products of cycle classes - II 作为周期类乘积的交替群 - II
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-26 DOI: 10.1007/s10801-024-01305-2
Harish Kishnani, Rijubrata Kundu, Sumit Chandra Mishra
{"title":"Alternating groups as products of cycle classes - II","authors":"Harish Kishnani, Rijubrata Kundu, Sumit Chandra Mishra","doi":"10.1007/s10801-024-01305-2","DOIUrl":"https://doi.org/10.1007/s10801-024-01305-2","url":null,"abstract":"<p>Given integers <span>(k,lge 2)</span>, where either <i>l</i> is odd or <i>k</i> is even, let <i>n</i>(<i>k</i>, <i>l</i>) denote the largest integer <i>n</i> such that each element of <span>(A_n)</span> is a product of <i>k</i> many <i>l</i>-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that <span>(lfloor frac{2kl}{3} rfloor le n(k,l)le lfloor frac{2kl}{3}rfloor +1)</span> [Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008]. It is known that the conjecture holds when <span>(k=2,3,4)</span>. Moreover, it is also true when <span>(3mid l)</span>. In this article, we determine the exact value of <i>n</i>(<i>k</i>, <i>l</i>) when <span>(3not mid l)</span> and <span>(kge 5)</span>. As an immediate consequence, we get that <span>(n(k,l)&lt;lfloor frac{2kl}{3}rfloor )</span> when <span>(kge 5)</span> and <span>(3not mid l)</span>, which shows that the above conjecture is not true in general. In fact in this case, the difference between the exact value of <i>n</i>(<i>k</i>, <i>l</i>) and the conjectured value grows linearly in terms of <i>k</i>. Our results complete the determination of <i>n</i>(<i>k</i>, <i>l</i>) for all values of <i>k</i> and <i>l</i>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"41 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969121","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
Anti-dendriform algebras, new splitting of operations and Novikov-type algebras 反树枝形代数、新拆分运算和诺维科夫型代数
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-26 DOI: 10.1007/s10801-024-01303-4
{"title":"Anti-dendriform algebras, new splitting of operations and Novikov-type algebras","authors":"","doi":"10.1007/s10801-024-01303-4","DOIUrl":"https://doi.org/10.1007/s10801-024-01303-4","url":null,"abstract":"<h3>Abstract</h3> <p>We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti-<span> <span>({mathcal {O}})</span> </span>-operators and anti-Rota–Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of <em>q</em>-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006966","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
Resistance diameters and critical probabilities of Cayley graphs on irreducible complex reflection groups 不可还原复反射群上 Cayley 图的电阻直径和临界概率
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-25 DOI: 10.1007/s10801-024-01302-5
Maksim Vaskouski, Hanna Zadarazhniuk
{"title":"Resistance diameters and critical probabilities of Cayley graphs on irreducible complex reflection groups","authors":"Maksim Vaskouski, Hanna Zadarazhniuk","doi":"10.1007/s10801-024-01302-5","DOIUrl":"https://doi.org/10.1007/s10801-024-01302-5","url":null,"abstract":"<p>We consider networks on minimal Cayley graphs of irreducible complex reflection groups <i>G</i>(<i>m</i>, <i>p</i>, <i>n</i>). We show that resistance diameters of these graphs have asymptotic <span>(Theta (1/n))</span> as <span>(nrightarrow infty )</span> under fixed <i>m</i>, <i>p</i>. Non-trivial lower and upper asymptotic bounds for critical probabilities of percolation for there appearing a giant connected component have been obtained.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947189","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
Multi-part cross-intersecting families 多部分交叉族
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-25 DOI: 10.1007/s10801-024-01301-6
Yuanxiao Xi, Xiangliang Kong, Gennian Ge
{"title":"Multi-part cross-intersecting families","authors":"Yuanxiao Xi, Xiangliang Kong, Gennian Ge","doi":"10.1007/s10801-024-01301-6","DOIUrl":"https://doi.org/10.1007/s10801-024-01301-6","url":null,"abstract":"<p>Let <span>({mathcal {A}}subseteq {[n]atopwithdelims ()a})</span> and <span>({mathcal {B}}subseteq {[n]atopwithdelims ()b})</span> be two families of subsets of [<i>n</i>], we say <span>({mathcal {A}})</span> and <span>({mathcal {B}})</span> are cross-intersecting if <span>(Acap Bne emptyset )</span> for all <span>(Ain {mathcal {A}})</span>, <span>(Bin {mathcal {B}})</span>. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"35 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969217","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
Common transversals and complements in abelian groups 无边群中的公共横截面和补集
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-24 DOI: 10.1007/s10801-024-01299-x
{"title":"Common transversals and complements in abelian groups","authors":"","doi":"10.1007/s10801-024-01299-x","DOIUrl":"https://doi.org/10.1007/s10801-024-01299-x","url":null,"abstract":"<h3>Abstract</h3> <p>Given a finite abelian group <em>G</em> and cyclic subgroups <em>A</em>, <em>B</em>, <em>C</em> of <em>G</em> of the same order, we find necessary and sufficient conditions for <em>A</em>, <em>B</em>, <em>C</em> to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups, we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions. </p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947110","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
Using mixed dihedral groups to construct normal Cayley graphs and a new bipartite 2-arc-transitive graph which is not a Cayley graph 利用混合二面群构建正常的 Cayley 图和一种新的非 Cayley 图的双方形 2 弧形传递图
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-23 DOI: 10.1007/s10801-024-01300-7
Daniel R. Hawtin, Cheryl E. Praeger, Jin-Xin Zhou
{"title":"Using mixed dihedral groups to construct normal Cayley graphs and a new bipartite 2-arc-transitive graph which is not a Cayley graph","authors":"Daniel R. Hawtin, Cheryl E. Praeger, Jin-Xin Zhou","doi":"10.1007/s10801-024-01300-7","DOIUrl":"https://doi.org/10.1007/s10801-024-01300-7","url":null,"abstract":"<p>A <i>mixed dihedral group</i> is a group <i>H</i> with two disjoint subgroups <i>X</i> and <i>Y</i>, each elementary abelian of order <span>(2^n)</span>, such that <i>H</i> is generated by <span>(Xcup Y)</span>, and <span>(H/H'cong Xtimes Y)</span>. In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph <span>(textrm{Cay}(H,(Xcup Y){setminus }{1}))</span> is equal to <span>(Hrtimes A(H,X,Y))</span>, where <i>A</i>(<i>H</i>, <i>X</i>, <i>Y</i>) is the setwise stabiliser in <span>({{,textrm{Aut},}}(H))</span> of <span>(Xcup Y)</span>. We use this criterion to resolve a question of Li et al. (J Aust Math Soc 86:111-122, 2009), by constructing a 2-arc-transitive normal cover of order <span>(2^{53})</span> of the complete bipartite graph <span>({{textbf {K}}}_{16,16})</span> and prove that it is <i>not</i> a Cayley graph.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947111","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 note on “Largest independent sets of certain regular subgraphs of the derangement graph” 关于 "错乱图的某些规则子图的最大独立集 "的说明
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-23 DOI: 10.1007/s10801-024-01304-3
Yuval Filmus, Nathan Lindzey
{"title":"A note on “Largest independent sets of certain regular subgraphs of the derangement graph”","authors":"Yuval Filmus, Nathan Lindzey","doi":"10.1007/s10801-024-01304-3","DOIUrl":"https://doi.org/10.1007/s10801-024-01304-3","url":null,"abstract":"<p>Let <span>(D_{n,k})</span> be the set of all permutations of the symmetric group <span>(S_n)</span> that have no cycles of length <i>i</i> for all <span>(1 le i le k)</span>. In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph <span>(text {Cay}(S_n,D_{n,k}))</span> is equal to the set of all the largest independent sets in the derangement graph <span>(text {Cay}(S_n,D_{n,1}))</span>, provided <i>n</i> is sufficiently large in terms of <i>k</i>. We give a simpler proof that holds for all <i>n</i>, <i>k</i> and also applies to the alternating group.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"174 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947020","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
Compact hyperbolic Coxeter four-dimensional polytopes with eight facets 具有八个面的紧凑双曲考斯特四维多面体
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-20 DOI: 10.1007/s10801-023-01279-7
Jiming Ma, Fangting Zheng
{"title":"Compact hyperbolic Coxeter four-dimensional polytopes with eight facets","authors":"Jiming Ma, Fangting Zheng","doi":"10.1007/s10801-023-01279-7","DOIUrl":"https://doi.org/10.1007/s10801-023-01279-7","url":null,"abstract":"<p>In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"114 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927665","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 direct sum of q-matroids q 个矩阵的直接和
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-17 DOI: 10.1007/s10801-023-01283-x
Michela Ceria, Relinde Jurrius
{"title":"The direct sum of q-matroids","authors":"Michela Ceria, Relinde Jurrius","doi":"10.1007/s10801-023-01283-x","DOIUrl":"https://doi.org/10.1007/s10801-023-01283-x","url":null,"abstract":"<p>For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for <i>q</i>-matroids, the <i>q</i>-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the <i>q</i>-analogue of matroid union we come to a definition of the direct sum of <i>q</i>-matroids. As a motivation for this definition, we show it has some desirable properties.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139754301","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 rank augmentation theorem for rank three string C-group representations of the symmetric groups 对称群的三阶弦 C 群表示的秩增强定理
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-02-16 DOI: 10.1007/s10801-023-01291-x
Julie De Saedeleer, Dimitri Leemans, Jessica Mulpas
{"title":"A rank augmentation theorem for rank three string C-group representations of the symmetric groups","authors":"Julie De Saedeleer, Dimitri Leemans, Jessica Mulpas","doi":"10.1007/s10801-023-01291-x","DOIUrl":"https://doi.org/10.1007/s10801-023-01291-x","url":null,"abstract":"<p>We give a rank augmentation technique for rank three string C-group representations of the symmetric group <span>(S_n)</span> and list the hypotheses under which it yields a valid string C-group representation of rank four thereof.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"64 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139754419","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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