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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 Q3 MATHEMATICS Pub Date : 2024-02-26 DOI: 10.1007/s10801-024-01305-2
Harish Kishnani, Rijubrata Kundu, Sumit Chandra Mishra

Given integers (k,lge 2), where either l is odd or k is even, let n(kl) denote the largest integer n such that each element of (A_n) is a product of k many l-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that (lfloor frac{2kl}{3} rfloor le n(k,l)le lfloor frac{2kl}{3}rfloor +1) [Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008]. It is known that the conjecture holds when (k=2,3,4). Moreover, it is also true when (3mid l). In this article, we determine the exact value of n(kl) when (3not mid l) and (kge 5). As an immediate consequence, we get that (n(k,l)<lfloor frac{2kl}{3}rfloor ) when (kge 5) and (3not mid l), which shows that the above conjecture is not true in general. In fact in this case, the difference between the exact value of n(kl) and the conjectured value grows linearly in terms of k. Our results complete the determination of n(kl) for all values of k and l.

给定整数(k,lge 2), 其中l为奇数或k为偶数,让n(k, l)表示最大整数n,使得(A_n)的每个元素都是k多个l循环的乘积。赫佐格(M. Herzog)、卡普兰(G. Kaplan)和列夫(A. Lev)猜想 (lfloor frac{2kl}{3}n(k,l)le lfloor frac{2kl}{3}rfloor +1)[Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008].众所周知,当 (k=2,3,4)时,猜想成立。此外,当 (3mid l) 时猜想也成立。在本文中,我们将确定当(3,3,4)和(k,5)时n(k,l)的精确值。作为一个直接的结果,我们得到了当(kge 5) 和(3not mid l) 时的(n(k,l)<lfloor frac{2kl}{3}rfloor ),这表明上述猜想在一般情况下是不正确的。事实上,在这种情况下,n(k, l)的精确值与猜想值之间的差值是以k为单位线性增长的。我们的结果完成了对所有 k 和 l 值的 n(k,l)的确定。
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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 Q3 MATHEMATICS Pub Date : 2024-02-26 DOI: 10.1007/s10801-024-01303-4

Abstract

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- ({mathcal {O}}) -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 q-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.

摘要 我们引入了反树形代数的概念,作为拆分关联性的一种新方法。反树枝形代数的特征是:两个乘法分别给出它们的左乘法算子和右乘法算子,这些算子的对立面在这两个乘法的和上定义了一个双模结构,而这个双模结构是联立的。由于树枝形代数的特征与此密切类似,因此使用这个术语是有道理的。我们引入了关联代数上的反({mathcal {O}}) 算子和反罗塔-巴克斯特算子的概念来解释反树状代数。特别是,在具有非生成交换康内斯环的关联代数上存在兼容的反树形代数结构。一个重要的观察结果是,某些亚类的树枝形代数和反树枝形代数之间存在q代数的对应关系。其直接结果是,我们给出了诺维科夫型树状布拉斯的概念,作为树状布拉斯的诺维科夫布拉斯的类似物,它与诺维科夫布拉斯的关系与树状布拉斯和前李布拉斯的关系是一致的。最后,我们扩展提供了一个引入反树枝形代数类似物概念的一般框架,它诠释了一种新的运算拆分。
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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 Q3 MATHEMATICS Pub Date : 2024-02-25 DOI: 10.1007/s10801-024-01302-5
Maksim Vaskouski, Hanna Zadarazhniuk

We consider networks on minimal Cayley graphs of irreducible complex reflection groups G(mpn). We show that resistance diameters of these graphs have asymptotic (Theta (1/n)) as (nrightarrow infty ) under fixed m, p. Non-trivial lower and upper asymptotic bounds for critical probabilities of percolation for there appearing a giant connected component have been obtained.

我们考虑了不可还原复反射群 G(m,p,n)的最小 Cayley 图上的网络。我们证明,在固定的 m、p 条件下,这些图的阻力直径具有渐近的 (Theta (1/n)) as (nrightarrow infty )。
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引用次数: 0
Multi-part cross-intersecting families 多部分交叉族
IF 0.8 3区 数学 Q3 MATHEMATICS Pub Date : 2024-02-25 DOI: 10.1007/s10801-024-01301-6
Yuanxiao Xi, Xiangliang Kong, Gennian Ge

Let ({mathcal {A}}subseteq {[n]atopwithdelims ()a}) and ({mathcal {B}}subseteq {[n]atopwithdelims ()b}) be two families of subsets of [n], we say ({mathcal {A}}) and ({mathcal {B}}) are cross-intersecting if (Acap Bne emptyset ) for all (Ain {mathcal {A}}), (Bin {mathcal {B}}). 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.

让({mathcal {A}}subseteq {[n]atopwithdelims()a})和({mathcal {B}}subseteq {[n]atopwithdelims()b})是[n]的两个子集族、如果对于所有的(A in {mathcal {A}}),(B in {mathcal {B}}),(Acap Bne emptyset )都是交叉的,我们就说({mathcal {A}})和({mathcal {B}})是交叉的。在本文中,我们将研究多部分环境下的交叉相交族。通过描述顶点变换图的独立集及其直接乘积,我们确定了最大尺寸的多部分交叉族的大小和结构。这概括了希尔顿(J Lond Math Soc 15(2):369-376, 1977)和弗兰克尔-托胡希(Frankl-Tohushige)(J Comb Theory Ser A 61(1):87-97, 1992)关于单部分交叉族的结果。
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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 Q3 MATHEMATICS Pub Date : 2024-02-24 DOI: 10.1007/s10801-024-01299-x

Abstract

Given a finite abelian group G and cyclic subgroups A, B, C of G of the same order, we find necessary and sufficient conditions for A, B, C 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.

摘要 给定一个有限无秩群 G 和 G 的同阶循环子群 A、B、C,我们发现 A、B、C 的余集有必要条件和充分条件。对于任意数量的循环子群,我们给出了存在共同补集的充分条件。此外,在存在共同横截面的几种情况下,我们提供了具体的构造。
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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 Q3 MATHEMATICS Pub Date : 2024-02-23 DOI: 10.1007/s10801-024-01300-7
Daniel R. Hawtin, Cheryl E. Praeger, Jin-Xin Zhou

A mixed dihedral group is a group H with two disjoint subgroups X and Y, each elementary abelian of order (2^n), such that H is generated by (Xcup Y), and (H/H'cong Xtimes Y). In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph (textrm{Cay}(H,(Xcup Y){setminus }{1})) is equal to (Hrtimes A(H,X,Y)), where A(HXY) is the setwise stabiliser in ({{,textrm{Aut},}}(H)) of (Xcup Y). 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 (2^{53}) of the complete bipartite graph ({{textbf {K}}}_{16,16}) and prove that it is not a Cayley graph.

混合二面群是一个群 H,它有两个互不相交的子群 X 和 Y,每个子群都是阶为 (2^n) 的初等无常群,这样 H 由 (Xcup Y) 和 (H/H'cong Xtimes Y) 生成。在本文中,我们给出了一个充分条件,即 Cayley 图的自(textrm{Cay}(H、(Xcup Y){setminus }{1})) 等于 (Hrtimes A(H,X,Y)),其中 A(H,X,Y)是 (Xcup Y) 的 ({{,textrm{Aut},}}(H)) 中的集合稳定器。我们利用这个标准解决了 Li 等人(J Aust Math Soc 86:111-122, 2009)提出的一个问题,即构造了完整双向图 ({{textbf {K}}}_{16,16}) 的一个阶为 (2^{53}) 的 2-arc-transitive normal cover,并证明它不是一个 Cayley 图。
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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 Q3 MATHEMATICS Pub Date : 2024-02-23 DOI: 10.1007/s10801-024-01304-3
Yuval Filmus, Nathan Lindzey

Let (D_{n,k}) be the set of all permutations of the symmetric group (S_n) that have no cycles of length i for all (1 le i le k). In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph (text {Cay}(S_n,D_{n,k})) is equal to the set of all the largest independent sets in the derangement graph (text {Cay}(S_n,D_{n,1})), provided n is sufficiently large in terms of k. We give a simpler proof that holds for all nk and also applies to the alternating group.

让 (D_{n,k}) 是对称组 (S_n) 的所有排列的集合,这些排列在所有 (1 le i le k) 条件下都没有长度为 i 的循环。在上面提到的论文中,Ku、Lau 和 Wong 证明,只要 n 对 k 来说足够大,那么 Cayley 图 (text {Cay}(S_n,D_{n,k})) 中所有最大独立集的集合等于 derangement 图 (text {Cay}(S_n,D_{n,1})) 中所有最大独立集的集合。我们给出了一个更简单的证明,它对所有 n、k 都成立,并且同样适用于交替群。
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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 Q3 MATHEMATICS Pub Date : 2024-02-20 DOI: 10.1007/s10801-023-01279-7
Jiming Ma, Fangting Zheng

In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.

在本文中,我们获得了具有八个面的紧凑双曲考斯特四维多面体的完整分类。
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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 Q3 MATHEMATICS Pub Date : 2024-02-17 DOI: 10.1007/s10801-023-01283-x
Michela Ceria, Relinde Jurrius

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 q-matroids, the q-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 q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.

对于经典矩阵,直接求和是用现有矩阵生成新矩阵的最直接方法之一。本文定义了 q-matroids(matroids 的 q-analogue )的直接和。正如我们将试图说服读者的那样,这种方法远没有经典方法那么简单。通过使用亚模函数和矩阵联合的 q-analogue ,我们得出了 q 个矩阵的直接和的定义。作为这个定义的动机,我们将证明它具有一些理想的性质。
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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 Q3 MATHEMATICS Pub Date : 2024-02-16 DOI: 10.1007/s10801-023-01291-x
Julie De Saedeleer, Dimitri Leemans, Jessica Mulpas

We give a rank augmentation technique for rank three string C-group representations of the symmetric group (S_n) and list the hypotheses under which it yields a valid string C-group representation of rank four thereof.

我们给出了对称群 (S_n)的秩为三的弦 C 群表示的秩增强技术,并列出了它能产生秩为四的有效弦 C 群表示的假设。
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引用次数: 0
期刊
Journal of Algebraic Combinatorics
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