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The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups 对称群上一些非正态Cayley图的第二大特征值
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-08 DOI: 10.1016/j.jcta.2025.106097
Yuxuan Li, Binzhou Xia, Sanming Zhou
A Cayley graph on the symmetric group <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg="si4.svg"><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg="si5.svg"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg="si6.svg"><mml:mrow><mml:mi mathvariant="normal">Cay</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> on <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg="si7.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi
如果对称群Sn上的Cayley图的严格第二大特征值(即严格小于度的最大特征值)通过Sn的标准表示获得,则称其具有Aldous性质。对于1≤r<;k<n,设C(n,k;r)为Sn移动{1,…,r}中每一点的k个环的集合。最近,Siemons and Zalesski(2022)[26]提出了一个猜想,该猜想等价于对于任意n≥5且1≤r<;k<n,具有连接集C(n,k;r)的Sn上的非正态Cayley图Cay(Sn,C(n,k;r))具有Aldous性质。通过求解这个猜想,我们证明了除(i) (n,k,r)=(6,5,1)或(ii) n为奇数,k =n−1,且1≤r<;n2外,所有图都具有Aldous性质。在此过程中,我们确定了Sn的所有不可约表示,这些表示可以实现Cay(Sn,C(n,n - 1;r))的严格第二大特征值以及该图的最小特征值。
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For &lt;mml:math altimg=\"si2.svg\"&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;≤&lt;/mml:mo&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"&gt;&lt;&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"&gt;&lt;&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:math&gt;, let &lt;mml:math altimg=\"si267.svg\"&gt;&lt;mml:mi&gt;C&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo&gt;;&lt;/mml:mo&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:math&gt; be the set of &lt;ce:italic&gt;k&lt;/ce:italic&gt;-cycles of &lt;mml:math altimg=\"si1.svg\"&gt;&lt;mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt; moving every point in &lt;mml:math altimg=\"si4.svg\"&gt;&lt;mml:mo stretchy=\"false\"&gt;{&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mo&gt;…&lt;/mml:mo&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;}&lt;/mml:mo&gt;&lt;/mml:math&gt;. Recently, Siemons and Zalesski (2022) &lt;ce:cross-ref ref&gt;[26]&lt;/ce:cross-ref&gt; posed a conjecture which is equivalent to saying that for any &lt;mml:math altimg=\"si5.svg\"&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;≥&lt;/mml:mo&gt;&lt;mml:mn&gt;5&lt;/mml:mn&gt;&lt;/mml:math&gt; and &lt;mml:math altimg=\"si2.svg\"&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;≤&lt;/mml:mo&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"&gt;&lt;&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"&gt;&lt;&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:math&gt; the nonnormal Cayley graph &lt;mml:math altimg=\"si6.svg\"&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"normal\"&gt;Cay&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;C&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo&gt;;&lt;/mml:mo&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:math&gt; on &lt;mml:math altimg=\"si1.svg\"&gt;&lt;mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt; with connection set &lt;mml:math altimg=\"si267.svg\"&gt;&lt;mml:mi&gt;C&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo&gt;;&lt;/mml:mo&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:math&gt; has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) &lt;mml:math altimg=\"si7.svg\"&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"22 1","pages":"106097"},"PeriodicalIF":1.1,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 geometry of intersecting codes and applications to additive combinatorics and factorization theory 交码几何及其在加性组合学和分解理论中的应用
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106023
Martino Borello , Wolfgang Schmid , Martin Scotti
Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicities) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted 2-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.
交码是线性码,其中每两个非零码字都有非平凡的交支持。在本文中,我们通过证明非退化相交码对应于射影空间中不包含在两个超平面中的点集(具有多重性)来扩展这类码的理论。这种对应关系允许使用几何参数来演示属性并提供相交代码的构造。我们改进了现有的边界长度,并提供了短相交码的显式结构。最后,推广编码理论与有限阿贝尔群的组合不变量Davenport常数理论之间的联系,给出了加权2-wise Davenport常数的新的渐近界。然后,这些界给出了代数整数环和相关结构的因数分解的结果。
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引用次数: 0
A central limit theorem for a card shuffling problem 洗牌问题的中心极限定理
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-04-03 DOI: 10.1016/j.jcta.2025.106048
Shane Chern , Lin Jiu , Italo Simonelli
Given a positive integer n, consider a permutation of n objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set uniformly at random until only one object is left. In this paper, we investigate the distribution of Xn, the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value E[Xn], the variance Var[Xn], and higher central moments, and show that Xn satisfies a central limit theorem.
给定一个正整数n,考虑随机均匀选择的n个对象的排列。在这种排列中,我们收集由按升序排列的连续数字组成的最大子序列,称为块。然后合并每个块,在所有合并之后,这个新集合的元素被重新标记,从1到当前的元素数。我们继续均匀随机地排列和合并这个新集合,直到只剩下一个对象。在本文中,我们研究了Xn的分布,即该过程结束所需的排列数。特别地,我们找到了均值E[Xn]、方差Var[Xn]和高中心矩的显式渐近表达式,并证明了Xn满足中心极限定理。
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引用次数: 0
Separable elements and splittings in Weyl groups of type B B型Weyl群的可分离元素与分裂
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106021
Ming Liu, Houyi Yu
Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair (X,Y) of subsets of the symmetric group Sn, the multiplication map X×YSn is a splitting (i.e., a length-additive bijection) of Sn if and only if X is the generalized quotient of Y and Y is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type B.
Weyl群中的可分离元素是对称群中著名的可分离置换的推广。Gaetz和Gao证明了对于对称群Sn的任意子集对(X,Y),当且仅当X是Y的广义商,Y是由可分离元素生成的右弱阶主低阶理想时,乘法映射X×Y→Sn是Sn的分裂(即长度加性双射)。他们推测这个结果可以推广到所有有限Weyl群。本文用禁止模式对所有可分和最小不可分有符号置换进行了分类,并证实了B型Weyl群的Gaetz和Gao猜想。
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引用次数: 0
Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances 满足Griesmer界或具有最优最小距离的二进制自正交码
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-02-28 DOI: 10.1016/j.jcta.2025.106027
Minjia Shi , Shitao Li , Tor Helleseth , Jon-Lark Kim
The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing the Solomon-Stiffler codes. As a result, we reduce a problem with an infinite number of cases to a finite number of cases. Second, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code of a binary self-orthogonal code. Using such a characterization, we completely determine the exact value of dso(n,7), where dso(n,k) denotes the largest minimum distance among all binary self-orthogonal [n,k] codes.
本文的目的是双重的。首先,我们利用solomon - stiff码刻画了满足Griesmer界的二进制自正交码的存在性。因此,我们将一个有无限种情况的问题简化为有限种情况。其次,通过考虑二进制自正交码的残差码,给出了证明某些二进制自正交码不存在的一般方法。利用这样的表征,我们完全确定了dso(n,7)的精确值,其中dso(n,k)表示所有二进制自正交[n,k]码之间的最大最小距离。
{"title":"Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances","authors":"Minjia Shi ,&nbsp;Shitao Li ,&nbsp;Tor Helleseth ,&nbsp;Jon-Lark Kim","doi":"10.1016/j.jcta.2025.106027","DOIUrl":"10.1016/j.jcta.2025.106027","url":null,"abstract":"<div><div>The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing the Solomon-Stiffler codes. As a result, we reduce a problem with an infinite number of cases to a finite number of cases. Second, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code of a binary self-orthogonal code. Using such a characterization, we completely determine the exact value of <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi><mi>o</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mn>7</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi><mi>o</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> denotes the largest minimum distance among all binary self-orthogonal <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> codes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106027"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 bijection related to Bressoud's conjecture 与布雷苏德的猜想有关的一个问题
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106032
Y.H. Chen, Thomas Y. He
Bressoud introduced the partition function B(α1,,αλ;η,k,r;n), which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function B(α1,,αλ;η,k,r;n) in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon identities.
Bressoud引入了配分函数B(α1,…,αλ;η,k,r;n),用于统计具有一定差异条件的配分数。Bressoud以多重求和形式对配分函数B(α1,…,αλ;η,k,r;n)的生成函数提出了一个猜想。在这篇文章中,我们引入了一个与Bressoud猜想有关的双射。作为一个应用,我们给出了Göllnitz-Gordon身份的一个伴侣的证明。
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引用次数: 0
On de Bruijn rings and families of almost perfect maps 在德布鲁因环和几乎完美的地图上
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106030
Peer Stelldinger
De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape (m,n) appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd m=n{3,5,7} and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd m=n{3,5,7} as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications.
De Bruijn tori,或完美映射,是有限字母表中字母的二维周期性数组,其中每种可能的形状模式(m,n)在单个周期内恰好出现一次。虽然某些de Bruijn环面(例如奇数m=n∈{3,5,7}和偶数字母大小的平方环面)的存在性仍未得到解决,但在位置编码等应用中,次完美映射通常是足够的。这些地图捕获了大量的模式,每种模式最多出现一次。虽然以前生成这种次完美映射的方法只覆盖了可能模式的一小部分,但我们提出了一种生成任意模式形状和任意非素数字母表大小的几乎完美映射的构造方法,包括上面提到的奇数m=n∈{3,5,7}的方形环面,只要字母表大小是非素数。这是通过引入de Bruijn环,一个最小高度的次完美地图和几乎完美地图族概念的形式化来实现的。生成的次完美地图很容易解码,这使得它们非常适合位置编码应用。
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引用次数: 0
An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs Uq(sl2)的Clebsch-Gordan系数与Grassmann图的Terwilliger代数之间难以察觉的联系
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106028
Hau-Wen Huang
<div><div>The Clebsch–Gordan coefficients of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra <span><math><mi>H</mi></math></span> into <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. Let Ω denote a finite set of size <em>D</em> and <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></math></span> denote the power set of Ω. It is generally known that <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> supports a <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module. Let <em>k</em> denote an integer with <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi></math></span> and fix a <em>k</em>-element subset <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of Ω. By identifying <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> with <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi><mo>∖</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup><mo>⊗</mo><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup></math></span> this induces a <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module structure on <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> denoted by <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>. Pulling back via ♮ the <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> forms an <span><math><mi>H</mi></math></span>-module. When <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi><mo>−</mo><mn>1
U(sl2)的Clebsch-Gordan系数可以用Hahn多项式表示。这一现象可以用从通用哈恩代数H到U(sl2)⊗U(sl2)的代数同态解释。设Ω表示大小为D的有限集,2Ω表示Ω的幂集。众所周知,C2Ω支持U(sl2)-模块。设k为0≤k≤D的整数,固定Ω的一个k元素子集x0。通过将C2Ω与C2Ω∈x0⊗C2x0识别,在C2Ω上得到一个U(sl2)⊗U(sl2)模块结构,表示为C2Ω(x0)。通过缩回调,U(sl2)⊗U(sl2)-模C2Ω(x0)形成h模。当1≤k≤D−1时,h模C2Ω(x0)包涵Johnson图J(D,k)关于x0的Terwilliger代数。这个结果将两个看似无关的主题联系起来:U(sl2)的Clebsch-Gordan系数和Johnson图的Terwilliger代数。不幸的是,在q模拟的情况下,有些步骤失效了。本文通过绕弯路,成功地揭示了Uq(sl2)的Clebsch-Gordan系数与Grassmann图的Terwilliger代数之间不易察觉的联系。
{"title":"An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs","authors":"Hau-Wen Huang","doi":"10.1016/j.jcta.2025.106028","DOIUrl":"10.1016/j.jcta.2025.106028","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The Clebsch–Gordan coefficients of &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; into &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Let Ω denote a finite set of size &lt;em&gt;D&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; denote the power set of Ω. It is generally known that &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; supports a &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-module. Let &lt;em&gt;k&lt;/em&gt; denote an integer with &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and fix a &lt;em&gt;k&lt;/em&gt;-element subset &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of Ω. By identifying &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; this induces a &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-module structure on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; denoted by &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Pulling back via ♮ the &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-module &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; forms an &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-module. When &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106028"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Stirling permutation codes. II 斯特林排列码。2
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-07-11 DOI: 10.1016/j.jcta.2025.106093
Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh
In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types A and B descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several e-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.
在斯特林多项式的背景下,Gessel和Stanley引入了斯特林排列,在过去的几十年里引起了广泛的关注。近年来,我们引入了Stirling排列码,并提供了大量的等分布结果作为应用。本研究的目的是进一步分析斯特林排列码。首先,我们导出了A型和B型下降统计量在高八面体群上的联合分布的展开式,并得到了涉及其系数多项式零点的交错性质。接下来,我们证明了高八面体群中的符号置换与斯特林置换之间的紧密联系。我们还研究了三元二阶欧拉多项式和上升平台多项式的统一推广。利用Stirling排列码,给出了8变量多项式和17变量多项式的展开式,其中蕴涵了若干e正展开式,并阐明了若干统计量之间的联系。我们的结果推广了Bóna、Chen-Fu、Dumont、Haglund-Visontai、Janson和Petersen的结果。
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引用次数: 0
A symmetry on weakly increasing trees and multiset Schett polynomials 弱递增树和多集Schett多项式的对称性
IF 0.9 2区 数学 Q2 MATHEMATICS Pub Date : 2025-07-01 Epub Date: 2025-01-20 DOI: 10.1016/j.jcta.2025.106010
Zhicong Lin , Jun Ma
By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma–Mansour–Wang–Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin–Ma–Ma–Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of “even-degree nodes on odd levels” and “odd-degree nodes” on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial γ-positivity of the multiset Schett polynomials, a result which implies both the symmetry and the unimodality of these polynomials.
通过考虑递增树中节点的度数和水平的奇偶性,得到了Jacobi椭圆函数泰勒展开式系数的一种新的组合解释。作为这一新解释的一个应用,我们解决了Ma-Mansour-Wang-Yeh的一个猜想。Lin-Ma-Ma-Zhou统一了增长树和平面树的概念,在多集上引入了弱增长树。建立了弱递增树上“奇阶上偶数度节点”和“奇阶节点”的对称联合分布,将Schett多项式推广到多集。给出了这种对称的组合证明和代数证明,以及几个相关的有趣结果。此外,通过在树上引入群作用,我们证明了多集Schett多项式的偏γ正性,这一结果暗示了这些多项式的对称性和单模性。
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引用次数: 0
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Journal of Combinatorial Theory Series A
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