A characterization of Q-polynomial association schemes

On few-class $Q$-polynomial association schemes: feasible parameters and nonexistence results

We present the tables of feasible parameters of primitive $3$-class $Q$-polynomial association schemes and $4$- and $5$-class $Q$-bipartite association schemes (on up to $2800$, $10000$, and $50000$ vertices, respectively), accompanied by a number of nonexistence results for such schemes obtained by analysing triple intersection numbers of putative open cases.

Classification of partially metric Q-polynomial association schemes with $m_1 = 4$

We classify the Q-polynomial association schemes with $m_{1} = 4$ which are partially metric with respect to the nearest neighbourhood relation. An association scheme is partially metric with respect to a relation $R_1$ if the scheme graph of $R_2$ is exactly the distance-2 graph of the scheme graph of $R_1$ under a certain ordering of the relations.

m-距离正则图及其与多元P-多项式结合方案的关系

m-distance-regular graphs and their relation to multivariate P-polynomial association schemes

该研究定义m-距离正则图概念，证明其为多元P-多项式结合方案提供图论解释，还探究两类对象的精细结构与约束，建立(α,β)型二元P-多项式方案和2-距离正则图的关联。

Characterizations of regularity for certain $Q$-polynomial association schemes

It is shown that linked systems of symmetric designs with $a_1^*=0$ and mutually unbiased bases (MUB) are triply regular association schemes. In this paper, we characterize triple regularity of linked systems of symmetric designs by its Krein number. And we prove that maximal MUB carries a quadruply regular association scheme and characterize the quadruple regularity of MUB by its parameter.

非二元Johnson方案与由衰减空间得到的结合方案的二元$Q$-多项式结构

Bivariate $Q$-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces

本文通过显式计算Krein参数，证明非二元Johnson结合方案与衰减空间结合方案均为二元$Q$-多项式，完全解除了此前研究的参数限制，推进了高秩多变量$Q$-多项式结合方案的研究。

A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes

We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius-Schur indicators. This situation is similar to the pair of the dihedral group and the quaternion group of order 8. We also determine the structures of adjacency algebras of them over the rational number field.

The Multiplicities of a Dual-thin Q-polynomial Association Scheme

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.

Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes

Suzuki (1998) showed that an imprimitive Q-polynomial association scheme with first multiplicity at least three is either Q-bipartite, Q-antipodal, or with four or six classes. The exceptional case with four classes has recently been ruled out by Cerzo and Suzuki (2009). In this paper, we show the nonexistence of the last case with six classes. Hence Suzuki's theorem now exactly mirrors its well-known counterpart for imprimitive distance-regular graphs.

On relative $t$-designs in polynomial association schemes

Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nice…

On the isomorphism of certain primitive $Q$-polynomial not $P$-polynomial association schemes

In 2011, Penttila and Williford constructed an infinite new family of primitive $Q$-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space $H(3,q^2)$, $q$ even, with respect to a symplectic polar space $W(3,q)$ embedded in it. In a private communication to Penttila and Williford, H.~Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of $\mathrm{PGL}(2,q^2)$, $q$ even, on a non-degenerate conic of $\mathrm{P…

Three-class association schemes from cyclotomy

We give three constructions of three-class association schemes as fusion schemes of the cyclotomic scheme, two of which are primitive

Every 4-equivalenced association scheme is Frobenius

For a positive integer $k$, we say that an association scheme $(Ω,S)$ is $k$-equivalenced if each non-diagonal element of $S$ has valency $k$. $k$-equivalenced is weaker than pseudocyclic. It is known that every $k$-equivalenced association scheme is Frobenius when $k=2,3$ and every $4$-equivalenced association scheme is pseudocyclic. In this paper, we will show that every $4$-equivalenced association scheme is Frobenius.

Spherical embeddings of symmetric association schemes in 3-dimensional Euclidean space

We classify the symmetric association schemes with faithful spherical embedding in 3-dimensional Euclidean space. Our result is based on previous research on primitive association schemes with $m_1 = 3$.

On circular-arc graphs with association schemes

In this paper, we give a characterization of the class of all circular-arc graphs whose schemes are association. Moreover, all association schemes which are the scheme of a circular-arc graph are characterized, specially it is proved that they are Schurian.

Reconstructing a generalized quadrangle from the Penttila-Williford $4-$class association scheme

Penttila and Williford constructed a $4-$class association scheme from a generalized quadrangle with a doubly subtended subquadrangle. We show that an association scheme with appropriate parameters and satisfying some assumption about maximal cliques must be the Penttila-Williford scheme.

Wreath products and projective system of non Schurian association schemes

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, w…

Infinite family of nonschurian separable association schemes

An infinite family of nonschurian separable association schemes is constructed.

导出三个部分几何设计的秩4自对偶结合方案的构造

Construction of rank $4$ self-dual association schemes inducing three partial geometric designs

针对B. Xu刻画的导出三个部分几何设计的秩4自对偶结合方案，本文给出其在阿贝尔2-群上作为Schur环的构造。

Association schemes all of whose symmetric fusion schemes are integral

In this paper we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.

基于衰减空间的结合方案的二元 $P$- 和 $Q$-多项式结构

Bivariate $P$- and $Q$-polynomial structures of the association schemes based on attenuated spaces

本文研究基于衰减空间的结合方案的二元 $P$/ $Q$-多项式结构，借助对偶 $q$-Hahn 与仿射 $q$-Krawtchouk 多项式的邻接关系揭示其双谱代数和子成分代数，并与非二元 Johnson 方案比较。

Pseudocyclic association schemes and strongly regular graphs

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantee the existe…

关于均匀Higman关联方案

On uniform Higmanian association schemes

本文研究了具有两个非平凡抛物线的Higman关联方案的均匀条件，给出了必要充分条件并提供了Cayley方案的例子。

Functoriality of Bose-Mesner algebras and profinite association schemes

We show that taking the set of primitive idempotents of commutative association schemes is a functor from the category of commutative association schemes with surjective morphisms to the category of finite sets with surjective partial functions. We then consider projective systems of commutative association schemes consisting of surjections (which we call profinite association schemes), for which Bose-Mesner algebra is defined, and describe a Delsarte theory on such schemes. This is another method for generalizing association schemes to those on infinite sets, related with the approach by Barg…

On intriguing sets of the Penttila-Williford association scheme

We investigate intriguing sets of an association scheme introduced by Penttila and Williford (2011) that was the basis for their construction of primitive cometric association schemes that are not $P$-polynomial nor the dual of a $P$-polynomial scheme. In particular, we give examples and characterisation results for the four types of intriguing sets that arise in this scheme.

On Wreath Products of One-Class Association Schemes

We give a full description of the algebraic structures of the Bose-Mesner algebra and Terwilliger algebra of the wreath product of one-class association schemes.

Partially metric association schemes with a multiplicity three

An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the Möbius-Kantor graph, the Nauru graph, and the Foster g…

Pseudocyclic and non-amorphic fusion schemes of the cyclotomic association schemes

We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanov's conjecture by Ikuta and Munemasa [15].

关于Tatra关联方案的可分离性

On separability of Tatra association schemes

本文证明了所有Tatra关联方案都是2-可分离的，即由其2维交数张量确定其同构类。

有向强正则图与可除设计图来自Tatra关联方案

Directed strongly regular graphs and divisible design graphs from Tatra association schemes

本文提出通过Tatra关联方案的新参数构造有向强正则图和可除设计图，并研究其关联方案及同构性。

从幺正群出发的三元组关联方案

Association Scheme on Triples from the Unitary Group

本文研究了从有限幺正群作用于各向同性线所生成的三元组关联方案的结构与性质。

Association schemes with at most two nonlinear irreducible characters and applications to finite groups

An irreducible character $χ$ of an association scheme is called nonlinear if the multiplicity of $χ$ is greater than $1$. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given.

Full automorphism groups of association schemes based on attenuated spaces

The set of subspaces with a given dimension in an attenuated space has a structure of a symmetric association scheme, which is a generalization of both Grassmann schemes and bilinear forms schemes. In [K. Wang, J. Guo, F. Li, Association schemes based on attenuated space, European J. Combin. 31 (2010) 297--305], its intersection numbers were computed. In this paper, we determine its full automorphism group.

关于关联方案的紧致泛化

On a Compact Generalization of Association Schemes

本文提出紧致关联方案的概念，将其与超群框架结合，并通过谐波分析视角拓展了关联方案理论。

Four-class Skew-symmetric Association Schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, …

每个非对称4类结合方案都可由一个有向图生成

Every nonsymmetric $4$-class association scheme can be generated by a digraph

该研究在特定假设下刻画了恰含一对非对称关系的交换结合方案的有向图生成条件，并应用该结论证明所有非对称4类结合方案都可由有向图生成，拓展了3类结合方案生成相关结论。

非二元Johnson方案的双变量$Q$-多项式结构

A bivariate $Q$-polynomial structure for the non-binary Johnson scheme

针对非二元Johnson方案，本文证明其在部分参数下具备双变量$Q$-多项式结构，结合已有的$P$-多项式结构得到一类双变量正交多项式的双谱性质，并研究了相关代数结构。

由一般酉群在迷向向量上的作用得到的结合方案

Association schemes obtained from the action of the general unitary group on isotropic vectors

本文研究了一般酉群在有限酉空间迷向向量集合上传递作用得到的无限族结合方案，计算了所有非平凡交换结合方案的参数并确定了其特征表。

On the Godsil -- Higman necessary condition for equitable partitions of association schemes

In the monograph 'Association schemes', C. Godsil derived a necessary condition for equitable partitions of association schemes and noticed that it could be used to show that certain equitable partitions do not exist. In this short note, we show that, in fact, this condition is not stronger than the well-known Lloyd theorem.

On schurian fusions of the association scheme of a Galois affine plane of prime order

The schurian fusions of the association scheme of a Galois affine plane of prime order are completely identified.

Duality of translation association schemes coming from certain actions

Translation association schemes are constructed from actions of finite groups on finite abelian groups satisfying certain natural conditions. It is also shown that the mere existence of maps from finite groups to themselves sending each element in their groups to its `adjoint' entails the self-duality of the constructed association schemes. Many examples of these, including Hamming scheme and sesquilinear forms schemes, are provided. This con- struction is further generalized to show the duality of the association schemes coming from actions of two finite groups on the same finite abelian grou…

A new semidirect product for association schemes

We propose a new definition for the semidirect product of association schemes, generalizing work of Bang, Hirasaka, and Song. We define an action of one scheme on another, and show how one can use such an action to construct all semidirect products satisfying a certain technical condition.

A two-fold cover of strongly regular graphs with spreads and association schemes of class five

We consider imprimitive association schemes of class four which are two-fold covers of strongly regular graphs with spreads. It will be shown that a two-fold cover of a strongly regular graph with a spread provides a five class fission scheme of the imprimitive scheme of class four.

携带局部关联方案的Roux方案

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。

关于拟薄Schurian结合方案的Terwilliger代数

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。

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.

Non-symmetric class $2$ association schemes obtained by doubling of skew-Hadamard matrices are non-schurian

We can obtain a non-symmetric class $2$ association scheme by a skew-Hadamard matrix. We begin with a skew-Hadamard matrix of order $n$, construct a skew-Hadamard matrix of order $2n$ by doubling construction, and a non-symmetric class $2$ association scheme of order $2n-1$. We will show that the association scheme obtained in this way never be schurian if $n$ is greater than or equal to $8$.

Extended double covers of non-symmetric association schemes of class $2$

In this paper, we give a method to construct non-symmetric association schemes of class $3$ from non-symmetric association schemes of class $2$. This construction is a non-symmetric analogue of the construction of Taylor graphs as an antipodal double cover of a complete graph. We also mention how our construction interact with doubling introduced by Pasechnik.

Group-case commutative association schemes and their character tables

Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect known group-case primitive commutative association schemes, and * compute their character tables over the last twenty years. The construction of their character tables are important first step for a systematic study of such association schemes and towards the classification of those schemes. In this talk, we briefly survey the progress made in this direction …

On tight $4$-designs in Hamming association schemes

We complete the classification of tight $4$-designs in Hamming association schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight $4$-design in $H(n,q)$ and analyze its triple intersection numbers to conclude the non-existence in all open cases.

Modular Terwilliger algebras of association schemes

We define modular Terwilliger algebras of association schemes, Terwilliger algebras over a positive characteristic field, and consider basic properties. We give a condition for the modular Terwilliger algebra to be non-semisimple. We show that the dimension of a Terwilliger algebra of a Johnson scheme depends on the characteristic of the coefficient field. We also give some other examples.

Symmetric Bush-type generalized Hadamard matrices and association schemes

We define Bush-type generalized Hadamard matrices over abelian groups and construct symmetric Bush-type generalized Hadamard matrices over the additive group of finite field $\mathbb{F}_q$, $q$ a prime power. We then show and study an association scheme obtained from such generalized Hadamard matrices.

结合方案中的近因子分解

Near-factorizations in association schemes

本文研究结合方案中的 $\lambda$ 重 $(s,t)$ 近因子分解，建立基本性质与参数界，完全刻画 2 类方案，构造多种方案实例，并证明某些方案不存在此类分解。

Reconstructing a generalized quadrangle with a hemisystem from a $4-$class association scheme

In 2013, van Dam, Martin and Muzychuk constructed a cometric $Q-$ antipodal $4-$class association scheme from a GQ of order $(t^2,t)$, $t$ odd, which have a hemisystem. In this paper we characterize this scheme by its Krein array. The techniques which are used involve the triple intersection numbers introduced by Coolsaet and Jurišić.

Self-orthogonal codes from equitable partitions of association schemes

We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.