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Growth of bilinear maps II: bounds and orders 双线性映射的增长 II：边界和阶数
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-21 DOI: 10.1007/s10801-024-01336-9
Vuong Bui
{"title":"Growth of bilinear maps II: bounds and orders","authors":"Vuong Bui","doi":"10.1007/s10801-024-01336-9","DOIUrl":"https://doi.org/10.1007/s10801-024-01336-9","url":null,"abstract":"<p>A good range of problems on trees can be described by the following general setting: Given a bilinear map <span>(*:mathbb {R}^dtimes mathbb {R}^drightarrow mathbb {R}^d)</span> and a vector <span>(sin mathbb {R}^d)</span>, we need to estimate the largest possible absolute value <i>g</i>(<i>n</i>) of an entry over all vectors obtained from applying <span>(n-1)</span> applications of <span>(*)</span> to <i>n</i> instances of <i>s</i>. When the coefficients of <span>(*)</span> are nonnegative and the entries of <i>s</i> are positive, the value <i>g</i>(<i>n</i>) is known to follow a growth rate <span>(lambda =lim _{nrightarrow infty } root n of {g(n)})</span>. In this article, we prove that for such <span>(*)</span> and <i>s</i> there exist nonnegative numbers <span>(r,r')</span> and positive numbers <span>(a,a')</span> so that for every <i>n</i>, </p><span>$$begin{aligned} a n^{-r}lambda ^nle g(n)le a' n^{r'}lambda ^n. end{aligned}$$</span><p>While proving the upper bound, we actually also provide another approach in proving the limit <span>(lambda )</span> itself. The lower bound is proved by showing a certain form of submultiplicativity for <i>g</i>(<i>n</i>). Corollaries include a lower bound and an upper bound for <span>(lambda )</span>, which are followed by a good estimation of <span>(lambda )</span> when we have the value of <i>g</i>(<i>n</i>) for an <i>n</i> large enough.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146499","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 weak Lefschetz property and mixed multiplicities of monomial ideals 弱列夫谢茨性质和单项式理想的混合乘数
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-21 DOI: 10.1007/s10801-024-01337-8
Thiago Holleben
{"title":"The weak Lefschetz property and mixed multiplicities of monomial ideals","authors":"Thiago Holleben","doi":"10.1007/s10801-024-01337-8","DOIUrl":"https://doi.org/10.1007/s10801-024-01337-8","url":null,"abstract":"<p>Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes <span>(Delta )</span> such that the squarefree reduction of the Stanley–Reisner ideal of <span>(Delta )</span> has the WLP in degree 1 and characteristic zero. In this paper, we apply the connections between analytic spread of equigenerated monomial ideals, mixed multiplicities and birational monomial maps to give a sufficient and necessary condition for the squarefree reduction <span>(A(Delta ))</span> to satisfy the WLP in degree <i>i</i> and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of <span>(Delta )</span>, we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of <span>(A(Delta ))</span> in degree <i>i</i> in positive characteristics in terms of mixed multiplicities. Moreover, we extend Dao and Nair’s criterion to arbitrary monomial ideals in positive odd characteristics.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"4280 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146431","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
Higher Bruhat orders of types B and C B 类和 C 类更高的布鲁哈特阶数
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-17 DOI: 10.1007/s10801-024-01334-x
Vladimir I. Danilov, Alexander V. Karzanov, Gleb A. Koshevoy
{"title":"Higher Bruhat orders of types B and C","authors":"Vladimir I. Danilov, Alexander V. Karzanov, Gleb A. Koshevoy","doi":"10.1007/s10801-024-01334-x","DOIUrl":"https://doi.org/10.1007/s10801-024-01334-x","url":null,"abstract":"<p>We propose versions of higher Bruhat orders for types <i>B</i> and <i>C</i>. This is based on a theory of higher Bruhat orders of type A and their geometric interpretations (due to Manin–Shekhtman, Voevodskii–Kapranov, and Ziegler), and on our study of the so-called symmetric cubillages of cyclic zonotopes.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"67 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061763","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
Nowhere-zero 3-flows in nilpotently vertex-transitive graphs 无顶点遍历图中的无处-零3流
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-16 DOI: 10.1007/s10801-024-01335-w
Junyang Zhang, Sanming Zhou
{"title":"Nowhere-zero 3-flows in nilpotently vertex-transitive graphs","authors":"Junyang Zhang, Sanming Zhou","doi":"10.1007/s10801-024-01335-w","DOIUrl":"https://doi.org/10.1007/s10801-024-01335-w","url":null,"abstract":"<p>We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061762","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
Counting and signed counting permutations by descent-based statistics 基于后裔统计的计数和符号计数排列
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-09 DOI: 10.1007/s10801-024-01330-1
Yao Dong, Zhicong Lin
{"title":"Counting and signed counting permutations by descent-based statistics","authors":"Yao Dong, Zhicong Lin","doi":"10.1007/s10801-024-01330-1","DOIUrl":"https://doi.org/10.1007/s10801-024-01330-1","url":null,"abstract":"<p>The original motivation of this paper was to find the context-free grammar for the joint distribution of peaks and valleys on permutations. Although such attempt was unsuccessful, we can obtain noncommutative symmetric function identities for the joint distributions of several descent-based statistics, including peaks, valleys and even/odd descents, on permutations via Zhuang’s generalized run theorem. Our results extend in a unified way several generating function formulas exist in the literature, including formulas of Carlitz and Scoville (Discrete Math 5:45–59, 1973; J Reine Angew Math 265:110–137, 1974), J. Combin. Theory Ser. A, 20: 336-356 (1976), Zhuang (Adv Appl Math 90:86–144, 2017), Pan and Zeng (Adv Appl Math 104:85–99, 2019; Discrete Math 346:113575, 2023). As applications of these generating function formulas, Wachs’ involution and Foata–Strehl action on permutations, we also investigate the signed counting of even and odd descents, and of descents and peaks, which provide two generalizations of Désarménien and Foata’s classical signed Eulerian identity.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939855","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 cyclic flats of a q-matroid q-matroid 的循环平面
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-09 DOI: 10.1007/s10801-024-01321-2
Gianira N. Alfarano, Eimear Byrne
{"title":"The cyclic flats of a q-matroid","authors":"Gianira N. Alfarano, Eimear Byrne","doi":"10.1007/s10801-024-01321-2","DOIUrl":"https://doi.org/10.1007/s10801-024-01321-2","url":null,"abstract":"<p>In this paper we develop the theory of cyclic flats of <i>q</i>-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a <i>q</i>-matroid and hence derive a new <i>q</i>-cryptomorphism. We introduce the notion of <span>(mathbb {F}_{q^m})</span>-independence of an <span>(mathbb {F}_q)</span>-subspace of <span>(mathbb {F}_q^n)</span> and we show that <i>q</i>-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"139 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939705","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
Intersection bodies of polytopes: translations and convexity 多边形的交点体：平移和凸性
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-09 DOI: 10.1007/s10801-024-01328-9
Marie-Charlotte Brandenburg, Chiara Meroni
{"title":"Intersection bodies of polytopes: translations and convexity","authors":"Marie-Charlotte Brandenburg, Chiara Meroni","doi":"10.1007/s10801-024-01328-9","DOIUrl":"https://doi.org/10.1007/s10801-024-01328-9","url":null,"abstract":"<p>We continue the study of intersection bodies of polytopes, focusing on the behavior of <i>IP</i> under translations of <i>P</i>. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of <span>(I(P+t))</span> can be extended to polynomials in variables <span>(tin mathbb {R}^d)</span> within each region of the arrangement. In dimension 2, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939854","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
Dihedral groups with the m-DCI property 具有 m-DCI 特性的二面群
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-08 DOI: 10.1007/s10801-024-01327-w
Jin-Hua Xie, Yan-Quan Feng, Young Soo Kwon
{"title":"Dihedral groups with the m-DCI property","authors":"Jin-Hua Xie, Yan-Quan Feng, Young Soo Kwon","doi":"10.1007/s10801-024-01327-w","DOIUrl":"https://doi.org/10.1007/s10801-024-01327-w","url":null,"abstract":"<p>A Cayley digraph <span>(textrm{Cay}(G,S))</span> of a group <i>G</i> with respect to a subset <i>S</i> of <i>G</i> is called a CI-digraph if for every Cayley digraph <span>(textrm{Cay}(G,T))</span> isomorphic to <span>(textrm{Cay}(G,S))</span>, there exists an <span>(alpha in textrm{Aut}(G))</span> such that <span>(S^alpha =T)</span>. For a positive integer <i>m</i>, <i>G</i> is said to have the <i>m</i>-DCI property if all Cayley digraphs of <i>G</i> with out-valency <i>m</i> are CI-digraphs. Li (European J Combin 18:655–665, 1997) gave a necessary condition for cyclic groups to have the <i>m</i>-DCI property, and in this paper, we find a necessary condition for dihedral groups to have the <i>m</i>-DCI property. Let <span>(textrm{D}_{2n})</span> be the dihedral group of order 2<i>n</i>, and assume that <span>(textrm{D}_{2n})</span> has the <i>m</i>-DCI property for some <span>(1 le mle n-1)</span>. It is shown that <i>n</i> is odd, and if further <span>(p+1le mle n-1)</span> for an odd prime divisor <i>p</i> of <i>n</i>, then <span>(p^2not mid n)</span>. Furthermore, if <i>n</i> is a power of a prime <i>q</i>, then <span>(textrm{D}_{2n})</span> has the <i>m</i>-DCI property if and only if either <span>(n=q)</span>, or <i>q</i> is odd and <span>(1le mle q)</span>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"44 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939556","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
Vertex stabilizers of locally s-arc transitive graphs of pushing up type 上推型局部s弧传图的顶点稳定器
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-08 DOI: 10.1007/s10801-024-01326-x
John van Bon, Chris Parker
{"title":"Vertex stabilizers of locally s-arc transitive graphs of pushing up type","authors":"John van Bon, Chris Parker","doi":"10.1007/s10801-024-01326-x","DOIUrl":"https://doi.org/10.1007/s10801-024-01326-x","url":null,"abstract":"<p>Suppose that <span>(Delta )</span> is a thick, locally finite and locally <i>s</i>-arc transitive <i>G</i>-graph with <span>(s ge 4)</span>. For a vertex <i>z</i> in <span>(Delta )</span>, let <span>(G_z)</span> be the stabilizer of <i>z</i> and <span>(G_z^{[1]})</span> the kernel of the action of <span>(G_z)</span> on the neighbours of <i>z</i>. We say <span>(Delta )</span> is of pushing up type provided there exist a prime <i>p</i> and a 1-arc (<i>x</i>, <i>y</i>) such that <span>(C_{G_z}(O_p(G_z^{[1]})) le O_p(G_z^{[1]}))</span> for <span>(z in {x,y})</span> and <span>(O_p(G_x^{[1]}) le O_p(G_y^{[1]}))</span>. We show that if <span>(Delta )</span> is of pushing up type, then <span>(O_p(G_x^{[1]}))</span> is elementary abelian and <span>(G_x/G_x^{[1]}cong X)</span> with <span>( textrm{PSL}_2(p^a)le X le mathrm{PGamma L}_2(p^a))</span>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"139 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939859","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
Möbius function of the subgroup lattice of a finite group and Euler characteristic 有限群子群晶格的莫比乌斯函数和欧拉特征
IF 0.8 3区 数学
Journal of Algebraic Combinatorics Pub Date : 2024-05-08 DOI: 10.1007/s10801-024-01329-8
Francesca Dalla Volta, Luca Di Gravina
{"title":"Möbius function of the subgroup lattice of a finite group and Euler characteristic","authors":"Francesca Dalla Volta, Luca Di Gravina","doi":"10.1007/s10801-024-01329-8","DOIUrl":"https://doi.org/10.1007/s10801-024-01329-8","url":null,"abstract":"<p>The Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup <i>G</i> of the general linear group <span>(textrm{GL}(n,q))</span> acting on the <i>n</i>-dimensional vector space <span>(V=mathbb {F}_q^n)</span>, where <span>(mathbb {F}_q)</span> is the finite field with <i>q</i> elements. We find a relation between this function and the Euler characteristic of two simplicial complexes <span>(Delta _1)</span> and <span>(Delta _2)</span>, the former raising from the lattice of the subspaces of <i>V</i>, the latter from the subgroup lattice of <i>G</i>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939555","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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