Q-polynomial Association Schemes Research Articles

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

The study of P-polynomial association schemes (distance-regular graphs) and Q-polynomial association schemes, and in particular P- and Q-polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of P- and Q-polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank P- and Q-polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi, and then followed by Bannai-Kurihara-Zhao-Zhu. The general theory and some explicit examples of families of higher rank (multivariate) P- and/or Q-polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate Q-polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the association schemes obtained from attenuated spaces are bivariate Q-polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study.

On association schemes generated by a relation or an idempotent

In this paper we study the polynomiality and co-polynomiality of association schemes. These two notions were introduced by Ito. We show that P-polynomial association schemes are co-polynomial and Q-polynomial association schemes are polynomial. We also determine the polynomiality and co-polynomiality for 3-class association schemes. Further we give examples of association schemes that are polynomial but not co-polynomial, or co-polynomial but not polynomial.

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.

Relative t-designs in Johnson association schemes for P-polynomial structure

Relative t-designs are defined in both P- and Q-polynomial association schemes. In this paper, we investigate relative t-designs in Johnson association schemes J(v, k) for P-polynomial structure. It is known that each nontrivial shell of J(v, k) is identified with the product of two smaller Johnson association schemes. We prove that relative t-designs in J(v, k) supported by one shell are equivalent to weighted $$\mathcal T$$ -designs in the shell (as product of association schemes) for $$\mathcal T=\{(t_1,t_2) \mid 0\le t_1,t_2\le t\}$$ . We study the existence problem of tight relative t-designs on one shell of J(v, k) for $$t=2,3$$ . We propose an algorithm to explicitly construct a family of non-trivial tight relative 2-designs. In addition, we obtain tight relative 3-designs for some special parameters.

The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

We prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes.

Design Theory from the Viewpoint of Algebraic Combinatorics

We give a survey on various design theories from the viewpoint of algebraic combinatorics. We will start with the following themes.(i)The similarity between spherical t-designs and combinatorial t-designs, as well as t-designs in Q-polynomial association schemes.(ii)Euclidean t-designs as a two-step generalization of spherical t-designs.(iii)Relative t-designs as a two-step generalization of t-designs in Q-polynomial association schemes, and the similarity with Euclidean t-designs.(iv)Fisher type lower bounds for the sizes of these designs as well as the classification problems of some of tight t-designs and/or tight relative t-designs. Our emphasis will be focused on the proposal to study relative t-designs, mostly tight relative t-designs, in known classical examples of P- and Q-polynomial association schemes. We relate our study with the representation theoretical aspect of the relevant association schemes and permutation groups, due to Charles Dunkl and Dennis Stanton and others. We propose several open problems, which seem to play a key role in this research direction. We also put emphasis on the future research directions in this research area and not on presenting the details of the established results on the study of design theory. In particular, we propose to study t-designs in each shell of these classical P- and Q-polynomial association schemes. In general, each shell is not Q-polynomial. It may be even non-commutative in some cases. The importance of the use of Terwilliger algebras in the study of relative t-designs in such association schemes will also be highlighted.

A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes

It is known that there is a close analogy between the two relations “Euclidean t-designs vs. spherical t-designs” and “relative t-designs in binary Hamming association schemes vs. combinatorial t-designs.” We first look at this analogy and survey the known results, putting emphasis on the study of tight relative t-designs in certain Q-polynomial association schemes. We then specifically study tight relative 2-designs on two shells in binary Hamming association schemes H(n, 2) and Johnson association schemes J(v, k). The purpose of this paper is to convince the reader that there is a rich theory even for these special cases and that the time is ripe to study tight relative t-designs more systematically for general Q-polynomial association schemes.

On the existence of tight relative 2-designs on binary Hamming association schemes

It is known that there is a close analogy between “Euclidean t-designs vs. spherical t-designs” and “Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs”. In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e≥2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-design on 2 shells in H(n,2) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n≤30. Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n≡6(mod8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.

Tight Relative 2- and 4-Designs on Binary Hamming Association Schemes

The concept of relative t-designs for Q-polynomial association schemes is due to Delsarte (Philips Res Rep 32:373---411, 1997). We formulate Fisher type lower bounds as well as the concept of tight relative 2- and 4-designs very explicitly for binary Hamming association schemes H(n, 2). Then we discuss some examples of tight relative 2- and 4-designs on H(n, 2) in this sense, in connection with the work of Woodall (Proc Lond Math Soc (3) 20:669---687, 1970), Enomoto-Ito-Noda (Osaka J Math 16:39---43, 1979), and others. The present paper is expected to be the first of a series of our studies of tight relative t-designs on Q-polynomial association schemes in a more general context.

An excess theorem for spherical 2-designs

We give an excess theorem for spherical 2-designs. This theorem is a dual version of the spectral excess theorem for graphs, which gives a characterization of distance-regular graphs, among regular graphs in terms of the eigenvalues and the excess. Here we give a characterization of Q-polynomial association schemes among spherical 2-designs.

New parameters of subsets in polynomial association schemes

We define new parameters, a zero interval and a dual zero interval, of subsets in P- or Q-polynomial association schemes. A zero interval of a subset in a P-polynomial association scheme is a successive interval index for which the inner distribution vanishes, and a dual zero interval of a subset in a Q-polynomial association scheme is a successive interval index for which the dual inner distribution vanishes. We derive bounds of the lengths of a zero interval and a dual zero interval using the degree and dual degree respectively, and show that a subset in a P-polynomial association scheme (resp. a Q-polynomial association scheme) having a large length of a zero interval (resp. a dual zero interval) induces a completely regular code (resp. a Q-polynomial association scheme). Moreover, we consider the spherical analogue of a dual zero interval.

A characterization of Q-polynomial association schemes

We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.

On spherical designs obtained from Q-polynomial association schemes

We characterize that the image of the embedding of the Q-polynomial association scheme into the first eigenspace by primitive idempotent E1 is a spherical t-design in terms of the Krein numbers. Furthermore, we show that the strengths of P- and Q-polynomial schemes as spherical designs are bounded by a constant. Copyright © 2011 John Wiley & Sons, Ltd. 19:167-177, 2011

New families of Q-polynomial association schemes

In this paper, we construct the first known infinite family of primitive Q-polynomial schemes which are not generated by distance-regular graphs. To construct these examples, we introduce the notion of a relative hemisystem of a generalized quadrangle with respect to a subquadrangle.

Designs in Product Association Schemes

A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte‘s bounds for t-designs in Q-polynomial association schemes.

Spectra of Bipartite P- and Q-Polynomial Association Schemes

Let Y=(X,{R i }0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A0, A1,…, A D of the associate matrices, and Q-polynomial with respect to the ordering E0, E1,…,E D of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv).

The Structure of Nonthin Irreducible T-modulesof Endpoint 1: Ladder Bases and Classical Parameters

Building on the work of Terwilliger, we find the structure of nonthin irreducible T-modules of endpoint 1 for P- and Q-polynomial association schemes with classical parameters. The isomorphism class of such a given module is determined by the intersection numbers of the scheme and one additional parameter which must be an eigenvalue for the first subconstituent graph. We show that these modules always have what we call a ladder basis, and find the structure explicitly for the bilinear, Hermitean, and alternating forms schemes.

Association Schemes with Multiple Q-polynomial Structures

It is well known that an association scheme \cX = (X,\{R_i\}_{0\leq i\leq d}) with k_1 > 2 has at most two P-polynomial structures. The parametrical condition for an association scheme to have two P-polynomial structures is also known. In this paper, we give a similar result for Q-polynomial association schemes. In fact, if d>5, then we obtain exactly the same parametrical conditions for the dual intersection numbers or Krein parameters.

Mixed block designs

Let Vi (i = 1, 2) be a set of size vi. Let D be a collection of ordered pairs (b1, b2) where bi is a ki-element subset of Vi. We say that D is a mixed t-design if there exist constants λ (j,j2), (0 ≤ ji ≤ ki, j1 + j2 ≤ t) such that, for every choice of a j1-element subset S1 of V1 and every choice of a j2-element subset S2 of V2, there exist exactly λ(j1,j2) ordered pairs (b1, b2) in D satisfying S1 ⊆ b1 and S2 ⊆ b2. In W. J. Martin [Designs in product association schemes, submitted for publication], Delsarte's theory of designs in association schemes is extended to products of Q-polynomial association schemes. Mixed t-designs arise as a particularly interesting case. These include symmetric designs with a distinguished block and α-resolvable balanced incomplete block designs as examples. The theory in the above-mentioned paper yields results on mixed t-designs analogous to those known for ordinary t-designs, such as the Ray-Chaudhuri/Wilson bound. For example, the analogue of Fisher's inequality gives |D| ≥ v1 + v2 − 1 for mixed 2-designs with Bose's condition on resolvable designs as a special case. Partial results are obtained toward a classification of those mixed 2-designs D with |D| = v1 + v2 − 1. The central result of this article is Theorem 3.1, an analogue of the Assmus–Mattson theorem which allows us to construct mixed (t + 1 − s)-designs from any t-design with s distinct block intersection numbers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:151–163, 1998

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes\(\mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) \) with \(\) are either bipartite or antipodal, i.e., intersection numbers satisfy either \(a_i = 0 \) for all\(i,{\text{or }}b_i = c_{d - i} \) for all \(i \ne [d/{\text{2}}] \). In this paper, we show that imprimitive \(Q \)-polynomial association schemes \(\mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) \) with \(\) are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either \(a_i^* = 0{\text{ for all }}i, or b_i^* = c_{d - 1}^* {\text{ for all }}i \ne [d/2] \).