Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

Balanced difference schemes in gas dynamics

Translated from Matematicheskie Modeli i Vychislitel'nye Metody, pp. 246–260, 1987.     

Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

A partial difference set (PDS) having parameters (n², r(n?1), n+r²?3r, r²?r) is called a Latin square type PDS, while a PDS having parameters (n², r(n+1), ?n+r²+3r, r² +r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2‐groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266‐282, 2009     

A construction of non-isomorphic amorphous association schemes from pseudo-cyclic association schemes

As a continuation of my paper [T. Fujisaki, A construction of amorphous association scheme from a pseudo-cyclic association scheme, Discrete Math. 285(1–3) (2004) 307–311], we show a construction of amorphous association scheme which is a fusion scheme of a direct product of two pseudo-cyclic association schemes with same first eigenmatrix. By using this construction, we can get at most three amorphous association scheme. We prove that if two pseudo-cyclic association scheme are non-isomorphic, then these three amorphous association schemes are mutually non-isomorphic.     

On threshold schemes from large sets

An anonymous (t, w)-threshold scheme is one of the schemes for secret sharing. Combinatorial designs and especially large sets of Steiner systems and of Steiner systems “with holes” have an important role in the design of perfect (t, w)-threshold schemes. In this article we investigate perfect (4, 4)-threshold schemes. We use large sets to form such systems with a large number of keys. In particular we construct the first known infinite families of large sets of H-designs with block size 4. © 1996 John Wiley & Sons, Inc.     

Fixed point sets through iteration schemes

We introduce the notion of a general fixed point iteration scheme to unify various fixed point iterations in the literatures, and extend the concept of virtual stability of a selfmap to an iteration scheme to obtain a connection, through an explicit retraction, between the convergence set of the scheme and its fixed point set. Moreover, we illustrate how to apply our results to obtain a new criterion for contractibility of the fixed point set of a certain quasi-nonexpansive selfmap.     

On quasi-thin association schemes

An association scheme is called quasi-thin if each of its basic relations has valency 1 or 2. A quasi-thin scheme is called Kleinian if its thin residue is the Klein four-group with respect to relational multiplication. It is proved that any Kleinian quasi-thin scheme arises from a near-pencil on 3 points, from an affine plane of order 2, or from a projective plane of order 2. The main result in this paper is that any non-Kleinian quasi-thin scheme is schurian and separable. We also construct an infinite family of Kleinian quasi-thin schemes which is neither schurian nor separable.     

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 investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

Tannaka-Krein duality for association schemes

A duality theorem is formulated for noncommutative association schemes. This duality theorem contains as special cases (1) the Delsarte-Tamaschke duality theorem (which was essentially obtained by Kawala in 1942) for commutative association schemes, and (2) the Tannaka-Krein duality theorem for arbitrary finite groups.     

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.     

t-designs in classical association schemes

Thet-designs in the eight infinite families of classical association schemes are characterized.Partially supported by a fellowship from the Sloan foundation and by NSF grant DMS: 8500958.     

Uniqueness of certain association schemes

We prove the uniqueness of the two association schemes which appear in recent work of Henry Cohn and others in connection with their study of universally optimal spherical codes in Euclidean spaces: one is the class 4 association scheme with 40 vertices in and the other one is the class 3 association scheme with 64 vertices in . We prove the uniqueness mainly by geometric considerations with some computational help.     

Schur indices of association schemes

In this paper we introduce and investigate the theory of Schur indices arising from simple components of the rational adjacency algebras of association schemes and investigate methods for computing these indices.     

Clifford theory for association schemes

Clifford theory of finite groups is generalized to association schemes. It shows a relation between irreducible complex characters of a scheme and a strongly normal closed subset of the scheme. The restriction of an irreducible character of a scheme to a strongly normal closed subset contains conjugate characters with same multiplicities. Moreover some strong relations are obtained.     

Balanced sets and circuits in a transversal space

The study of fully dependent sets (unions of circuits) has played a part in characterizing transversal spaces. In fact, the fully dependent sets satisfy |Δ(F)| = ?(F) in any deltoid representation, and it is with a consideration of this property that we begin the present paper. We study “balanced” sets and from our results draw conclusions about fully dependent sets and circuits in a transversal space. These include upper bounds for the number of circuits, and the result that a non-trivial transversal space can be neither a hereditary circuit space nor the dual of a geometric hereditary circuit space. The paper is reasonably self-contained; all unusual terms are defined as they are encountered.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

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 (Eur J Combin 31:1513–1519, 2010).     

On p-covalenced association schemes

Let (X,S) denote an association scheme where X is a finite set. For a prime p we say that (X,S) is p-covalenced (p-valenced) if every multiplicity (valency, respectively) of (X,S) is a power of p. In the character theory of finite groups Ito's theorem states that a finite group G has a normal abelian p-complement if and only if every character degree of G is a power of p. In this article we generalize Ito's theorem to p-valenced association schemes, i.e., a p-valenced association scheme (X,S) has a normal p-covalenced p-complement if and only if (X,S) is p-covalenced.