M. E. Muzychuk, I. N. Ponomarenko
The notion of pseudocyclic association scheme is generalized to the non-commutative case. It is proved that any pseudocyclic scheme the rank of which is much more than the valency is the scheme of a Frobenius group and is uniquely determined up to isomorphism by its intersection number array. An immediate corollary of this result is that any scheme of prime degree, valency $k$ and rank at least $k^4$ is schurian.
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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].
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…
The action of $PGL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ affords an association scheme, which was shown to be pseudocyclic in Hollmann's thesis in 1982. It was further conjectured in Hollmann's thesis that the orbital scheme of $PΓL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ is also pseudocyclic if $m$ is an odd prime. We confirm this conjecture in this paper. As a by-product, we obtain a class of Latin square type strongly regular graphs on nonprime-power number of points.
Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${\mathrm{PSL}}(2,q)$ and ${\mathrm{PΓL}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${\mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exc…
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.
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…
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, …
The group $PGL(2,q)$ has an embedding into $PGL(3,q)$ such that it acts as the group fixing a nonsingular conic in $PG(2,q)$. This action affords a coherent configuration $R(q)$ on the set $L(q)$ of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $R_{+}(q)$ and $R_{-}(q)$ to the sets $L_{+}(q)$ of secant lines and to the set $L_{-}(q)$ of exterior lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $R_{-}(q)$ is pseudocyclic. We further show that the coherent …
We give a sufficient condition for a non-commutative association scheme to have a fusion association scheme, and construct non-commutative association schemes from symmetric balanced generalized weighing matrices and generalized Hadamard matrices. We then apply the criterion to these non-commutative association schemes to obtain symmetric fusion 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.
We give three constructions of three-class association schemes as fusion schemes of the cyclotomic scheme, two of which are primitive
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 $m_1 = 3$.
We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.
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.
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.
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…
An infinite family of nonschurian separable association schemes is constructed.
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。
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.
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。
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…
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.
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.
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…
We give an equivalent condition of the $P$-polynomial property of symmetric association schemes.
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。
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。
We show an inequality involving the third largest or second smallest dual eigenvalues of $Q$-polynomial association schemes of class at least three. Also we characterize dual-tight $Q$-polynomial association schemes of class three. Our method is based on tridiagonal matrices and can be applied to distance-regular graphs as well.
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。
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.
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.
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.
Bivariate $P$-polynomial association schemes
本研究定义了作为P-多项式结合方案推广的$(α,β)$型双变量P-多项式结合方案，证明其等价于一组交参数条件，发现诸多高秩结合方案属此类，并类似定义了双变量Q-多项式结合方案。
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.
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。
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。
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。
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.
The schurian fusions of the association scheme of a Galois affine plane of prime order are completely identified.
In [H. Suzuki, Association schemes with multiple $Q$-polynomial structures, J. Algebraic Combin. 7 (1998) 181-196], Suzuki gave a classification of association schemes with multiple $Q$-polynomial structures, allowing for one exceptional case which has five classes. In this paper, we rule out the existence of this case. Hence Suzuki's theorem mirrors exactly the well-known counterpart for association schemes with multiple $P$-polynomial structures, a result due to Eiichi Bannai and Etsuko Bannai in 1980.
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…
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.
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.
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.
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。
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 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.
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 $8$.
We characterize that the image of the embedding of the $Q$-polynomial association scheme into eigenspace by primitive idempotent $E_1$ is a spherical $t$-design in terms of the Krein numbers. And we show that the strengths of $P$- and $Q$-polynomial schemes as spherical designs are bounded by constant.
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.
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 …
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.
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.
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.
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。
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.
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.
For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.
Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square type graphs and whose underlying set is a non-elementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes.
It is inevitable that the $L_{g}(s)$ association scheme with $g\geq 3, s\geq g+2$ is a pseudo-$L_{g}(s)$ association scheme. On the contrary, although $s^2$ treatments of the pseudo-$L_{g}(s)$ association scheme can form one $L_{g}(s)$ association scheme, it is not always an $L_{g}(s)$ association scheme. Mainly because the set of cardinality $s$, which contains two first-associates treatments of the pseudo-$L_{g}(s)$ association scheme, is non-unique. Whether the order $s$ of a Latin square $\mathbf{L}$ is a prime power or not, the paper proposes two new conditions in order to extend a $POL(s…
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