John S. Caughman, IV, Bruce E. Sagan
Let Y denote a symmetric association scheme which is Q-polynomial with respect to an ordering E_0,...,E_D of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities m_0,...,m_D is unimodal. We prove that if Y is dual-thin in the sense of Terwilliger, then the sequence of multiplicities satisfies m_i <= m_{i+1} and m_i <= m_{D-i} for i < D/2.
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In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic variety for which any suitable integral solution corresponds to a standard integral table algebra with integral multip…
On the Terwilliger algebras of quasi-thin Schurian association schemes
本文给出拟薄结合方案为三重传递的充要条件，并由此构造无穷多新的三重传递结合方案族。
An association scheme is called quasi-thin if the valency of each its basic relation is one or two. A quasi-thin scheme is Kleinian if the thin residue of it forms a Klein group with respect to the relation product. It is proved that any Kleinian scheme arises from near-pencil on~$3$ points, or affine or projective plane of order~$2$. The main result is that any non-Kleinian quasi-thin scheme a) is the two-orbit scheme of a suitable permutation group, and b) is characterized up to isomorphism by its intersection number array. An infinite family of Kleinian quasi-thin schemes for which neither …
In this article, we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of rank $2$.
In [3], Hanaki defined the Terwilliger algebras of association schemes over a commutative unital ring. In this paper, we call the Terwilliger algebras of association schemes over a field $\mathbb{F}$ the Terwilliger $\mathbb{F}$-algebras of association schemes and study the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. As main results, we determine the $\mathbb{F}$-dimensions, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. We also get some results with independent interes…
In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique trivial submodule to the case of the regular modules of modular adjacency algebras. We completely determine the $p$-transitive quasi-thin association schemes and the $p$-transitive association schemes with thin thin residue by their structure theory properties.
Quantum entropy and the Bannai--Ito multiplicity conjecture
本文通过量子熵的不等式证明了对称Q-多项式结合方案重数的对数凹性，解决了Bannai-Ito单峰猜想及Stanton相关猜想。
We construct a wide subcategory of the category of finite association schemes with a collection of desirable properties. Our subcategory has a first isomorphism theorem analogous to that of groups. Also, standard constructions taking schemes to groups (thin radicals and thin quotients) or algebras (adjacency algebras) become functorial when restricted to our category. We use our category to give a more conceptual account for a result of Hanaki concerning products of characters of association schemes; i.e. we show that the virtual representations of an association scheme form a module over the …
Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that any irreducible $T$-module $W$ is both thin and dual thin in the sense of Terwilliger. We produce two bases for $W$ and describe the action of $T$ on these bases. We prove that the isomorphism class of $W$ as a $T$-module is determined by two parameters, the dual endpoint and diameter of $W$. We find a recurrence which gives the multiplicities with which the i…
Structure of Terwilliger algebras of quasi-thin association schemes
本文研究拟薄结合方案的 Terwilliger 代数，证明其在任意域上为拟遗传胞腔代数，并在特征 $2$ 下刻画基本代数结构，从而完全确定其同调与表示论性质。
The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique we prove a new resu…
A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.
The log concavity of two graphical sequences
本文证明了任意图的大笛卡尔幂关于固定顶点的价序列及距离正则图的价序列均具有对数凹性，改进了已有结果，并推导出强正则图、二权码、完全正则码及Q-多项式结合方案重数序列的相关结论。
Roux schemes which carry association schemes locally
研究Roux方案的构造与特性，证明其邻域诱导关联方案的性质，并展示64等角线在C^8中的唯一性。
Classification of thin Jordan schemes
本文对最大秩到阶数比的乔丹方案进行分类，证明正则乔丹方案对应于环交替循环。
We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in the category algebra of the underlying category of a schemoid. In this paper, the equivalence between the categories of grouopids and that of thin association schemoids is established. Moreover linear extensions of schemoids are considered. A general theory of the Baues-Wirsching cohomology deduces a classification theorem for such extensions of a schemoid. We…
One may think of a $d$-class association scheme as a $(d+1)$-dimensional matrix algebra over $\mathbb{R}$ closed under Schur products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the Schur product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. For any association scheme we find a basis of $d+1$ idempotent matrices for the algebra. A cometric scheme is one whose idempotent basis…
Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an association scheme on triples (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and…
The Terwilliger algebras of doubly regular tournaments
本研究探讨对应双正则竞赛图的非对称秩3结合方案的Terwilliger代数性质，明确其薄不可约模与维数特征，验证了阶23内复Terwilliger代数可区分方案，而阶27时需用有理Terwilliger代数列表区分。
We first summarize the basic structure of the outer distribution module of a completely regular code. Then, employing a simple lemma concerning eigenvectors in association schemes, we propose to study the tightest case, where the indices of the eigenspace that appear in the outer distribution module are equally spaced. In addition to the arithmetic codes of the companion paper, this highly structured class includes other beautiful examples and we propose the classification of $Q$-polynomial completely regular codes in the Hamming graphs. A key result is Theorem 3.10 which finds that the $Q$-po…
Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. Consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. Such a pair is called a Leonard pair on $V$. In this paper we introduce a mild…
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