Commutative Association Schemes Research Articles

Difference sets in pseudocyclic association schemes

Wilson (1972) proved that for fixed positive integers d and r such that 2d divides r(r−1), there exists a subset Y of size r of the finite field of order q such that the r(r−1) nonzero differences occurring from Y are evenly distributed to the dth cyclotomic classes provided that q is sufficiently large relative to d and r. On the other hand, Kajiura, Matsumoto and Okuda (2019) introduced the concept of difference sets in commutative association schemes in their study on a quasi-Monte Carlo method to approximate integrations over the point sets. Then, Wilson's result implies the existence of difference sets in cyclotomic schemes. In this paper, we generalize Wilson's result into pseudocyclic commutative association schemes and show the existence of difference sets in pseudocyclic association schemes with sufficiently large numbers of points.

Self‐dual association schemes, fusions of Hamming schemes, and partial geometric designs

AbstractPartial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self‐dual association schemes. Then using the characterizations of self‐dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self‐dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2‐designs. In particular, for a primitive self‐dual symmetric association scheme of rank 4, if is a power of 3 and each of , , and induces a partial geometric design, then we will prove that is algebraically isomorphic to a fusion scheme of the Hamming scheme for some odd number .

The search for small association schemes with noncyclotomic eigenvalues

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 set for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. The main results of this paper show the eigenvalues of association schemes of rank 4 and nonsymmetric association schemes of rank 5 will always be cyclotomic. In the rank 5 cases, the results rely on calculations done by computer for Gröbner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.

Partial geometric designs arising from association schemes

The basic relations of an association scheme induce block designs in a natural way. There are many (commutative or non-commutative) association schemes in which some basic relations induce partial geometric designs or 2-designs. In this paper we study how the algebraic structures of association schemes determine partial geometric designs. We will first establish a necessary and sufficient condition under which some basic relations of an association scheme induce partial geometric designs. Then by using the representation theory of association schemes, we will show that the parameters of these partial geometric designs have some special properties. We will also use the characters of commutative association schemes to determine which basic relations induce partial geometric designs or 2-designs. Isomorphisms of partial geometric designs will also be discussed.

On the Brauer-Wielandt-Harada theorem for group-like regular association schemes

We consider a generalization of the Brauer-Wielandt-Harada Theorem to group-like regular association schemes. As an application, we give a necessary condition for commutative association schemes to be regular. Moreover, we derive the number of irreducible characters of multiplicity 1 from the product of all adjacency matrices and all valencies for a commutative regular association scheme.

On association schemes with multiplicities 1 or 2

Inspired by the work of Amitsur [1] on finite groups whose irreducible characters all have degree (multiplicity) 1 or 2, in this paper we study association schemes whose irreducible characters all have multiplicity 1 or 2. We will first show that the general case can be reduced to commutative association schemes. Then for commutative association schemes with multiplicities 1 or 2, we prove that their Krein parameters are all rational integers. Using automorphism groups of association schemes, we obtain a characterization and classification of those commutative association schemes all valencies and multiplicities of which are 1 or 2 in terms of Cayley schemes.

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.

Disjoint weighing matrices

The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search leads to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes.

On structures of quasi-thin table algebras and applications to association schemes

In this paper we develop a structure theory for a class of table algebras, i.e. the so-called quasi-thin table algebras. They are direct generalizations of quasi-thin association schemes. Although this class of table algebras is rather close to finite groups, its structural properties are quite different from those of finite groups. The thin radical and thin residue give rise to two group structures for every table algebra. They play the central role in our discussions. Applications to association schemes are also discussed, and some new results about quasi-thin association schemes are obtained. In particular, we will give necessary and sufficient conditions under which a commutative quasi-thin association scheme has multiplicities 1 or 2. We will need these results in the study of the characterization and classification of double quasi-thin association schemes in [22].

Commutative association schemes obtained from twin prime powers, Fermat primes, Mersenne primes

For prime powers q and q+ε where ε∈{1,2}, an affine resolvable design from Fq and Latin squares from Fq+ε yield a set of symmetric designs if ε=2 and a set of symmetric group divisible designs if ε=1. We show that these designs derive commutative association schemes, and determine their eigenmatrices.

Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes

We will give an equation on Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes.

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.

Balancedly splittable Hadamard matrices

Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.

CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS

Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$ . We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.

Tight [formula omitted]-designs on one shell of Johnson association schemes

Each nontrivial shell (i.e., subconstituent) of the Johnson association scheme J(v,k) is known to be a commutative association scheme which is the product of two smaller Johnson association schemes. The concept of t-designs in one shell of J(v,k) was naturally defined and studied by Martin in the context of mixed t-designs. The purpose of this paper is to try to push this study a bit further. First we give a direct proof of the theorem that if (Y,w) is a relative t-design in J(v,k) on p shells then the part in each shell must be a weighted (t−p+1)-design in the shell. In particular, it is a mixed (t−p+1)-design in one shell if the weight function is constant on each shell. We also present another approach that makes use of the Terwilliger algebra, based on the work of Tanaka. This result is essentially proved by Martin in 1998, but the proof there contains a small gap which is repaired in this paper. We also study the existence problems of tight 2-, 3- and 4-designs on one shell of Johnson association scheme J(v,k) with small parameters, say v≤1000, thereby expanding the range of search in the original paper of Martin in 1998.

Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

On the Connectivity of Graphs in Association Schemes

Let $(X,\mathcal{R})$ be a commutative association scheme and let $\Gamma=(X,R\cup R^\top)$ be a connected undirected graph where $R\in \mathcal{R}$. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of $\Gamma$ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two vertices $a,b\in X$ are called "twins" in $\Gamma$ if they have identical neighborhoods: $\Gamma(a)=\Gamma(b)$. We characterize twins in polynomial association schemes and show that, in the absence of twins, the deletion of any vertex and its neighbors in $\Gamma$ results in a connected graph. Using this and other tools, we find lower bounds on the connectivity of $\Gamma$, especially in the case where $\Gamma$ has diameter two.

On association schemes with multiplicities 1 or a prime p

ABSTRACTThe main result of this paper gives a classification of commutative association schemes, all irreducible characters of which have multiplicity 1 or a prime p.

An Assmus–Mattson theorem for codes over commutative association schemes

We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with $s$ classes). This in particular generalizes the Assmus-Mattson-type theorems for $\mathbb{Z}_4$-linear codes due to Tanabe (2003) and Shin, Kumar, and Helleseth (2004), as well as the original theorem by Assmus and Mattson (1969). The weights of a code are $s$-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining $t$-designs from the code involve concepts from polynomial interpolation in $s$ variables. The Terwilliger algebra is the main tool to establish our results.

Erdős–Ko–Rado for perfect matchings

A perfect matching of a complete graph K2n is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if F is family of intersecting perfect matchings of K2n, then |F|≤(2(n−1)−1)!! and if equality holds, then F=Fij where Fij is the family of all perfect matchings of K2n that contain some fixed edge ij. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family F is non-Hamiltonian, that is, m∪m′≇C2n for any m,m′∈F, then |F|≤(2(n−1)−1)!!. Our results make ample use of a symmetric commutative association scheme arising from the Gelfand pair (S2n,S2≀Sn). We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.