Journal of Combinatorial Theory Series A最新文献

英文中文

Erdős-Ko-Rado theorem for bounded multisets 有界多集的 Erdős-Ko-Rado 定理

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-03-14 DOI: 10.1016/j.jcta.2024.105888

Jiaqi Liao , Zequn Lv , Mengyu Cao , Mei Lu

<div>Let <math><mi>k</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi></math> be positive integers with <math><mi>k</mi><mo>⩾</mo><mn>2</mn></math>. A k-multiset of <math><msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mi>m</mi></mrow></msub></math> is a collection of k integers from the set <math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math> in which the integers can appear more than once but at most m times. A family of such k-multisets is called an intersecting family if every pair of k-multisets from the family have non-empty intersection. A finite sequence of real numbers <math><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> is said to be unimodal if there is some <math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math>, such that <math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⩽</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⩽</mo><mo>…</mo><mo>⩽</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⩽</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⩾</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⩾</mo><mo>…</mo><mo>⩾</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. Given <math><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi></math>, denote <math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math> as the coefficient of <math><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></math> in the generating function <math><msup><mrow><mo>(</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>ℓ</mi></mrow></msup></math>, where <math><mn>1</mn><mo>⩽</mo><mi>ℓ</mi><mo>⩽</mo><mi>n</mi></math>. In this paper, we first show that the sequence of <math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>)</mo></math> is unimodal. Then we use this as a tool to prove that the intersecting family in which every k-multiset contains a fixed element attains the maximum cardinality for <math><mi>n</mi><mo>⩾</mo><mi>k</mi><mo>+</mo><mrow><mo>⌈</mo><

设 k,m,n 是 k⩾2 的正整数。一个 [n]m 的 k 多集是从集合 {1,2,...,n}中选出的 k 个整数的集合，其中的整数可以出现不止一次，但最多出现 m 次。如果族中的每一对 k 多集都有非空的交集，那么这样的 k 多集族称为交集族。如果存在某个 k∈{1,2,...,n}，使得 a1⩽a2⩽...⩽ak-1⩽ak⩾ak+1⩾...⩾an，则称实数的有限序列 (a1,a2,...an) 为单模序列。给定 m,n,k，表示 Ck,ℓ 为 xk 在生成函数 (∑i=1mxi)ℓ 中的系数，其中 1⩽ℓ⩽n。在本文中，我们首先证明（Ck,1,Ck,2,...,Ck,n）序列是单峰的。然后，我们以此为工具证明，在 n⩾k+⌈k/m⌉ 的交集族中，每个 k 多集都包含一个固定元素，从而达到最大心数。在 m=1 和 m=∞ 的特殊情况下，我们的结果分别引出了著名的厄尔多斯-柯-拉多定理，以及 Meagher 和 Purdy [11] 所给出的该问题的无界多集版本。本文的主要结果可以看作是 Erdős-Ko-Rado 定理的有界多集版本。

引用次数: 0

Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture 广义海森伯变体的双元几何和广义沙雷西安-瓦克斯猜想

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-03-04 DOI: 10.1016/j.jcta.2024.105884

Young-Hoon Kiem , Donggun Lee

We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group $S_{n}$ on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the $S_{n}$ -representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.

我们引入广义海森堡变项并建立基本事实。我们证明了对称群 Sn 对海森堡变的同调的泰莫茨科作用扩展到广义海森堡变，并且广义海森堡变之间的自然形态保留了这一作用。通过分析广义海森堡变项间的自然形态和双映射，我们给出了沙雷西安-瓦克斯猜想的基本证明。此外，我们还提出了涉及广义海森堡变项的 Shareshian-Wachs 猜想的自然广义化，并给出了基本证明。作为副产品，我们提出了加权图的广义斯坦利-斯坦桥猜想。我们对广义海森堡变项的双向几何的研究，使我们能够通过明确的炸开序列或投影束映射，将它们修改成更简单的变项，如投影空间或包面变项。利用这一点，我们提供了两种算法来计算广义海森伯变项同调上的 Sn 代表。作为应用，我们计算了一些海森堡变项的低度同调上的表示。

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引用次数: 0

The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles 由循环生成的对称群上正态 Cayley 图的第二大特征值

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-03-01 DOI: 10.1016/j.jcta.2024.105885

Yuxuan Li, Binzhou Xia, Sanming Zhou

We study the normal Cayley graphs $Cay (S_{n}, C (n, I))$ on the symmetric group $S_{n}$ , where $I \subseteq {2, 3, \dots, n}$ and $C (n, I)$ is the set of all cycles in $S_{n}$ with length in I. We prove that the strictly second largest eigenvalue of $Cay (S_{n}, C (n, I))$ can only be achieved by at most four irreducible representations of $S_{n}$ , and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when I contains neither $n - 1$ nor n we know exactly when $Cay (S_{n}, C (n, I))$ has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of $S_{n}$ , and we obtain that $Cay (S_{n}, C (n, I))$ does not have the Aldous property whenever $n \in I$ . As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of $Cay (S_{n}, C (n, {k}))$ where $2 \leq k \leq n - 2$ .

我们研究了对称群 Sn 上的正则 Cayley 图 Cay(Sn,C(n,I))，其中 I⊆{2,3,...,n}，C(n,I) 是 Sn 中长度在 I 中的所有循环的集合。我们证明了 Cay(Sn,C(n,I)) 的严格第二大特征值最多只能由 Sn 的四个不可还原表示来实现，并进一步确定了该特征值在几种特殊情况下的多重性。作为一个推论，在 I 既不包含 n-1 也不包含 n 的情况下，我们可以准确地知道 Cay(Sn,C(n,I)) 何时具有阿尔多斯性质，即严格意义上的第二大特征值是由 Sn 的标准表示达到的，并且我们得到，只要 n∈I ，Cay(Sn,C(n,I)) 就不具有阿尔多斯性质。作为我们主要结果的另一个推论，我们证明了最近关于 Cay(Sn,C(n,{k}))第二大特征值的猜想，其中 2≤k≤n-2.

引用次数: 0

A short combinatorial proof of dimension identities of Erickson and Hunziker 埃里克森和亨兹克维度等式的简短组合证明

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-29 DOI: 10.1016/j.jcta.2024.105883

Nishu Kumari

In a recent paper (arXiv:2301.09744), Erickson and Hunziker consider partitions in which the arm–leg difference is an arbitrary constant m. In previous works, these partitions are called $(- m)$ -asymmetric partitions. Regarding these partitions and their conjugates as highest weights, they prove an identity yielding an infinite family of dimension equalities between ${gl}_{n}$ and ${gl}_{n + m}$ modules. Their proof proceeds by the manipulations of the hook content formula. We give a simple combinatorial proof of their result.

在最近的一篇论文（arXiv:2301.09744）中，埃里克森和亨兹克考虑了手脚差为任意常数 m 的分区。将这些分区及其共轭作为最高权重，他们证明了一个特性，即在 gln 和 gln+m 模块之间产生了一个无限维相等的系列。他们的证明是通过对勾股定理公式的操作进行的。我们给出了他们结果的简单组合证明。

引用次数: 0

On the deepest cycle of a random mapping 关于随机映射的最深循环

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-22 DOI: 10.1016/j.jcta.2024.105875

Ljuben Mutafchiev , Steven Finch

Let $T_{n}$ be the set of all mappings $T : {1, 2, \dots, n} \to {1, 2, \dots, n}$ . The corresponding graph of T is a union of disjoint connected unicyclic components. We assume that each $T \in T_{n}$ is chosen uniformly at random (i.e., with probability $n^{- n}$ ). The cycle of T contained within its largest component is called the deepest one. For any $T \in T_{n}$ , let $ν_{n} = ν_{n} (T)$ denote the length of this cycle. In this paper, we establish the convergence in distribution of $ν_{n} / \sqrt{n}$ and find the limits of its expectation and variance as $n \to \infty$ . For n large enough, we also show that nearly 55% of all cyclic vertices of a random mapping $T \in T_{n}$ lie in its deepest cycle and that a vertex from the longest cycle of T does not belong to its largest component with approximate probability 0.075.

设 Tn 是所有映射 T:{1,2,...,n}→{1,2,...,n} 的集合。T 的对应图是互不相连的单环部分的联合。我们假设每个 T∈Tn 都是均匀随机选择的（即概率为 n-n）。T 的最大分量所包含的循环称为最深循环。对于任意 T∈Tn，让 νn=νn(T) 表示这个循环的长度。在本文中，我们建立了 νn/n 分布的收敛性，并找到了其期望和方差随 n→∞ 的极限。对于足够大的 n，我们还证明了在随机映射 T∈Tn 的所有循环顶点中，有近 55% 的顶点位于其最深的循环中，并且 T 最长循环中的顶点不属于其最大分量的概率近似为 0.075。

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引用次数: 0

Two conjectures of Andrews, Merca and Yee on truncated theta series 安德鲁斯、梅尔卡和易关于截断θ级数的两个猜想

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-22 DOI: 10.1016/j.jcta.2024.105874

Shane Chern , Ernest X.W. Xia

In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function $M_{k} (n)$ to count the number of partitions of n in which k is the least integer that is not a part and there are more parts exceeding k than there are below k. In recent years, two conjectures concerning $M_{k} (n)$ on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large n whenever k is fixed.

安德鲁斯和梅尔卡在研究欧拉五边形数截断定理时，引入了一个分区函数 Mk(n)，用来计算 n 的分区数，其中 k 是不属于分区的最小整数，且超过 k 的分区数多于低于 k 的分区数。近年来，安德鲁斯、梅尔卡和易提出了关于截断θ数列 Mk(n) 的两个猜想。在本文中，我们证明了只要 k 固定不变，对于足够大的 n，这两个猜想都是真的。

引用次数: 0

Constructing generalized Heffter arrays via near alternating sign matrices 通过近交替符号矩阵构建广义赫夫特阵列

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-21 DOI: 10.1016/j.jcta.2024.105873

L. Mella , T. Traetta

<div>Let S be a subset of a group G (not necessarily abelian) such that <math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math> is empty or contains only elements of order 2, and let <math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math> and <math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. A generalized Heffter array GHA<math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> over G is an <math><mi>m</mi><mo>×</mo><mi>n</mi></math> matrix <math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math> such that: the i-th row (resp. j-th column) of A contains exactly <math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> (resp. <math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math>) nonzero elements, and the list <math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math> equals λ times the set <math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of A sums to zero (resp. a nonzero element), with respect to some ordering.In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (<math><mi>m</mi><mo>≠</mo><mi>n</mi></math>) and with less than n nonzero entries in each row. Furthermore, we build nonzero sum GHA<math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> over an arbitrary group G whenever S contains enough noninvolutions, th

设 S 是一个群 G 的子集（不一定是非良性的），使得 S∩-S 是空的或只包含阶数为 2 的元素，并且设 h=（h1,...,hm）∈Nm，k=（k1,...,kn ）∈Nn。G 上的广义赫夫特数组 GHASλ(m,n;h,k) 是一个 m×n 矩阵 A=(aij)，使得：A 的第 i 行（或第 j 列）恰好包含 hi（或 kj）个非零元素，且列表 {aij,-aij|aij≠0} 等于集合 S∪-S 的 λ 倍。在本文中，我们使用近交替符号矩阵来构建循环群上的零和（或非零和）GHA，它还具有简单的强性质。特别是，我们构建的零和简单 GHA，其行权重和列权重同余为 0 modulo 4。这一结果还提供了第一个矩形（m≠n）且每行非零条目少于 n 个的简单（经典）赫夫特数组无穷族。此外，只要 S 包含足够多的非卷积，我们就能在任意群 G 上建立非零和 GHASλ(m,n;h,k)，从而扩展了之前的非构造性结果，即对于 G 的某个子群 H，±S=G∖H。最后，我们描述了如何利用 GHA 来建立 Cayley 图（在不一定是无性的群上）到可定向曲面上的正交分解和双嵌套。

引用次数: 0

On the maximal number of elements pairwise generating the finite alternating group 关于成对生成有限交替群的元素的最大数目

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-14 DOI: 10.1016/j.jcta.2024.105870

Francesco Fumagalli , Martino Garonzi , Pietro Gheri

Let G be the alternating group of degree n. Let $ω (G)$ be the maximal size of a subset S of G such that $〈 x, y 〉 = G$ whenever $x, y \in S$ and $x \neq y$ and let $σ (G)$ be the minimal size of a family of proper subgroups of G whose union is G. We prove that, when n varies in the family of composite numbers, $σ (G) / ω (G)$ tends to 1 as $n \to \infty$ . Moreover, we explicitly calculate $σ (A_{n})$ for $n \geq 21$ congruent to 3 modulo 18.

设 G 是 n 阶交替群。设 ω(G) 是 G 的子集 S 的最大大小，当 x,y∈S 且 x≠y 时，使得〈x,y〉=G；设 σ(G) 是 G 的一族适当子群的最小大小，其联合是 G。我们证明，当 n 在合数族中变化时，σ(G)/ω(G) 随着 n→∞ 趋于 1。此外，我们还明确地计算了 n≥21 的 σ(An)与 3 的同余式 18 的同余式。

引用次数: 0

Most plane curves over finite fields are not blocking 有限域上的大多数平面曲线都不是阻塞的

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-09 DOI: 10.1016/j.jcta.2024.105871

Shamil Asgarli , Dragos Ghioca , Chi Hoi Yip

A plane curve $C \subset P^{2}$ of degree d is called blocking if every $F_{q}$ -line in the plane meets C at some $F_{q}$ -point. We prove that the proportion of blocking curves among those of degree d is $o (1)$ when $d \geq 2 q - 1$ and $q \to \infty$ . We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d \geq 3 p$ and $d, q \to \infty$ . Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of $F_{q}$ -roots of random polynomials, we find that the limiting distribution of the number of $F_{q}$ -points in the intersection of a random plane curve and a fixed $F_{q}$ -line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $F_{q}$ -points contained in a union of k lines for $k = 1, 2, \dots, q^{2} + q + 1$ .

如果平面中的每条 Fq 线都与 C 交于某个 Fq 点，则阶数为 d 的平面曲线 C⊂P2 称为阻塞曲线。我们证明，当 d≥2q-1 且 q→∞ 时，阻塞曲线在 d 阶曲线中所占比例为 o(1)。我们还证明，在较弱的条件 d≥3p 和 d,q→∞ 下，同样的结论也适用于光滑曲线。此外，随机平面曲线的光滑和阻塞两种情况被证明是渐近独立的。我们扩展了关于随机多项式 Fq 根数的经典结果，发现随机平面曲线与固定 Fq 线交点的 Fq 点数的极限分布是均值为 1 的泊松分布。我们还给出了阻塞曲线比例的明确公式，其中涉及 k=1,2,...q2+q+1 时 k 线结合处所含 Fq 点数的统计量。

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引用次数: 0

A Q-polynomial structure for the Attenuated Space poset Aq(N,M) 衰减空间正集 Aq(N,M) 的 Q 多项式结构

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-09 DOI: 10.1016/j.jcta.2024.105872

Paul Terwilliger

<div>The goal of this article is to display a Q-polynomial structure for the Attenuated Space poset <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math>. The poset <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math> is briefly described as follows. Start with an <math><mo>(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>)</mo></math>-dimensional vector space H over a finite field with q elements. Fix an M-dimensional subspace h of H. The vertex set X of <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math> consists of the subspaces of H that have zero intersection with h. The partial order on X is the inclusion relation. The Q-polynomial structure involves two matrices <math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math> with the following entries. For <math><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></math> the matrix A has <math><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></math>-entry 1 (if y covers z); <math><msup><mrow><mi>q</mi></mrow><mrow><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math> (if z covers y); and 0 (if neither of <math><mi>y</mi><mo>,</mo><mi>z</mi></math> covers the other). The matrix <math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> is diagonal, with <math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math>-entry <math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math> for all <math><mi>y</mi><mo>∈</mo><mi>X</mi></math>. By construction, <math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> has <math><mi>N</mi><mo>+</mo><mn>1</mn></math> eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with <math><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></math> eigenspaces. We show that <math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q-polynomial. We show that <math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of

本文的目的是展示衰减空间正集 Aq(N,M) 的 Q 多项式结构。正集 Aq(N,M) 简述如下。从有限域上具有 q 个元素的 (N+M) 维向量空间 H 开始。Aq(N,M)的顶点集 X 由与 h 有零交集的 H 子空间组成。Q 多项式结构涉及两个矩阵 A,A⁎∈MatX(C)，其条目如下。对于 y,z∈X，矩阵 A 有 (y,z) 项 1（如果 y 覆盖了 z）；qdimy（如果 z 覆盖了 y）；0（如果 y,z 都没有覆盖另一个）。矩阵 A⁎ 是对角线，对于所有 y∈X 都有 (y,y) 项 q-dimy。根据构造，A⁎ 有 N+1 个特征空间。根据构造，A 以（分块）三对角方式作用于这些特征空间。我们证明 A 是可对角的，有 2N+1 个特征空间。我们证明 A⁎ 以（块）三对角方式作用于这些特征空间。利用这一作用，我们证明 A 是 Q 多项式。我们证明 A、A⁎ 满足一对称为三对角关系的关系。我们考虑由 A,A⁎ 生成的 MatX(C) 子代数 T。我们证明，A,A⁎ 作为伦纳德对作用于每个不可还原 T 模块。

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