On relative designs in
polynomial association schemes
Abstract
Motivated by the similarities between the theory of spherical designs and that of designs in polynomial association schemes, we study two versions of relative designs, the counterparts of Euclidean designs for  and/or polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative designs, assuming that certain irreducible modules behave nicely. The two versions of relative designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.
Keywords: Relative design; Fisher type inequality; Terwilliger algebra
1 Introduction
Design theory is concerned with finding “good” finite sets that “approximate globally” their underlying spaces (often) having strong symmetry/regularity, such as the Euclidean space , the unit sphere , and the set of subsets of a given set. It has therefore a vast range of applications in various fields of science. See, e.g., [9, 3].
The similarities between the theories of spherical designs and combinatorial  designs are well known; cf. [15, 14, 19, 2]. Historically, the concept of spherical designs was introduced by Delsarte, Goethals, and Seidel [15] as a continuous analogue of that of designs in polynomial association schemes due to Delsarte [11, 12]. (Combinatorial  designs are precisely the designs in the Johnson scheme .) It was then generalized to the concept of Euclidean designs by Neumaier and Seidel [23], and Euclidean designs quickly became an active area of research; cf. [3]. Although the counterparts of Euclidean designs in the theory of polynomial association schemes were already defined and discussed to some extent by Delsarte [13] (cf. [4]) much earlier as relative designs, it seems that the theory of the latter has not been fully developed yet (except in the case of the binary Hamming scheme , in which case relative designs turn out to be equivalent to regular wise balanced designs). This paper is a contribution to this theory. Our discussions also include a concept of relative designs in general polynomial association schemes as well, following Delsarte and Seidel [16].
We refer the reader to [11, 6, 7, 19, 22, 10], etc., for the background on association schemes and some fundamental concepts. Throughout the paper, let be a (symmetric) class association scheme, and fix a base vertex . Let for . We call the shells of . Let be the vector space consisting of all the real valued functions on . In the following arguments we often identify with the vector space consisting of the real column vectors with coordinates indexed by .
We first introduce a concept of designs for general polynomial association schemes. Suppose that is polynomial with respect to the ordering . In the study of spherical/Euclidean designs in , we work with the vector space of polynomials in variables, in particular with the subspaces of homogeneous polynomials. For the polynomial scheme , it is natural to consider the following subspaces of . For every , we define by
In other words, if and only if lies on a geodesic between and in the corresponding distanceregular graph . Let . Then,
and we have the following direct sum decomposition of :
We now consider a (positive) weighted subset of , that is to say, a pair of a subset of and a function . Let , and let , for . We say that is supported by the union of shells. For any subspace of , we write .
Definition 1.1 (polynomial case).
A weighted subset of is a relative design of with respect to if
for every .
This definition is due to Delsarte and Seidel [16, Section 6] for the binary Hamming scheme . In this paper, we mostly consider the case for simplicity.
Theorem 1.2 ([16]).
Let be a relative design of a Hamming scheme with respect to in the sense of Definition 1.1. Let be the union of the shells which support . Then,
(1.1) 
Delsarte and Seidel [16] proved Theorem 1.2 only for , but their proof works for general . Theorem 1.2 also follows from Theorem 1.4 and Proposition 1.5 below. Recently, Xiang [35] succeeded in determining the right hand side of (1.1) explicitly for , which was left open in [16]. Namely, he proved
(1.2) 
under a reasonable additional condition which avoids the triviality. In this paper, we focus on generalizing (1.2) to other classes of polynomial association schemes (without necessarily reference to Theorem 1.2 itself). In Appendix A, we do, however, show that Theorem 1.2 is valid for dual polar schemes as well.
The concept of relative designs for polynomial association schemes was introduced by Delsarte [13]. We now recall the definition. Suppose that is polynomial with respect to the ordering of its primitive idempotents, and let be the column space of (). Then,
and we have the following orthogonal direct sum decomposition of :
Definition 1.3 (polynomial case).
A weighted subset of is a relative design of with respect to if
for every .
Bannai and Bannai [4] obtained the following Fisher type inequality for general polynomial association schemes:
Theorem 1.4 ([4]).
Let be a relative design of the polynomial scheme with respect to in the sense of Definition 1.3. Let be the union of the shells which support . Then,
(1.3) 
As in the case of (1.1), it was not easy to compute the right hand side of (1.3) explicitly. The initial attempt was made by Li, Bannai, and Bannai [21] for , but was unsuccessful in general. Then, this attempt lead Xiang to obtain a successful result in the general case for , as it is known that the two definitions of relative designs are essentially equivalent for . Namely, both definitions are shown to be equivalent to the geometric definition of relative designs coming from the structure of the regular semilattice associated with ; cf. [13]. The equivalence of Definition 1.1 for with the definition of regular wise balanced designs was pointed out by Delsarte and Seidel [16, Theorem 6.2], whereas the equivalence of Definition 1.3 for with the geometric definition of relative designs was established by Delsarte [13, Theorem 9.8] (see also [5]). However, we note that
Proposition 1.5.
If is a Hamming scheme , then for ,
(1.4) 
Proof.
Without loss of generality, we may suppose that and . Let . Note that has exactly nonzero entries, and let be the corresponding coordinates. Then, it is easy to see that is the characteristic function of the subset , which is known to be contained in ; see, e.g., [12, 26].^{1}^{1}1In Appendix B, we give a direct proof that belongs to , which does not use the theory of regular semilattices found in [12, 26]. Since both sides of (1.4) have the same dimension, we obtain the desired result. ∎
Thus, for , relative designs in the sense of Definition 1.1 are equivalent to relative designs in the sense of Definition 1.3. This observation seems to be new for for general . As is mentioned before, for , the result of Xiang [35] implies that the right hand side of (1.3) is also given explicitly by
(1.5) 
since in this case. In a private communication, Xiang extended his main result in [35] to general . Thus, the right hand side of (1.3) is also given explicitly as (1.5) for .
In this paper, we investigate to what extent the above results can be generalized to other  and/or polynomial association schemes. In Section 2, we derive sufficient conditions that (1.2) (resp. (1.5)) holds for a polynomial (resp. polynomial) association scheme (Theorems 2.3 and 2.7). These conditions can be readily checked for , so that we obtain different proofs of the results of Xiang mentioned above. Concerning (1.4), we first suspected that a similar result might hold for general (formally) selfdual  and polynomial association schemes, but it turns out that this is not the case in general. Indeed, in Section 3, we show that if is formally selfdual, polynomial (and thus polynomial), and satisfies , then must be a Hamming scheme , provided that (Theorem 3.2). All of these theorems are proved using the theory of the Terwilliger algebra [30, 31, 32]. See [28] for more applications of the Terwilliger algebra to design theory.
2 Computations of the Fisher type lower bounds
In this section and the next, we shall use some basic facts about the Terwilliger algebra. In this context, we shall work with the complex vector space instead of , but we note that the dimensions of the various subspaces in question do not change, as they are spanned by real vectors.
We use the following notation. For every , let be the characteristic function of the set . Let and be (fixed orderings of) the adjacency matrices and the primitive idempotents of , respectively. Let and be the diagonal matrices with diagonal entries and (, ). They form two bases of the dual Bose–Mesner algebra with respect to . When we assume that is polynomial (resp. polynomial), we understand that (resp. ) is the polynomial ordering (resp. polynomial ordering) and write (resp. ). The Terwilliger algebra with respect to is the subalgebra of the full matrix algebra generated by the Bose–Mesner algebra and the dual Bose–Mesner algebra. We note that is semisimple since it is closed under conjugatetransposition.
The endpoint, dual endpoint, diameter, and the dual diameter of an irreducible module are defined by , , , and , respectively.^{2}^{2}2In [30, 31, 32], , , , and are called the dual endpoint, endpoint, dual diameter, and the diameter of , respectively. The module is called thin (resp. dual thin) if (resp. ) (). There is a unique irreducible module with or up to isomorphism, that is to say, the primary module ; cf. [30, Lemma 3.6]. It is thin, dual thin, and has diameter and dual diameter both equal to . We call thin (resp. dual thin) with respect to if every irreducible module is thin (resp. dual thin).^{3}^{3}3We simply call thin (resp. dual thin) if it is thin (resp. dual thin) with respect to every base vertex . The next two lemmas will be freely used in our discussions.
Lemma 2.1 ([30, Lemma 3.9]).
Suppose that is polynomial. Let be an irreducible module and set , . Then, the following hold:

, where .

.

if .

If is thin, then ; in particular, is dual thin and .
Lemma 2.2 ([30, Lemma 3.12]).
Suppose that is polynomial. Let be an irreducible module and set , . Then, the following hold:

, where .

.

if .

If is dual thin, then ; in particular, is thin and .
We note that if is polynomial then
(2.1) 
for .
Theorem 2.3.
Suppose that is polynomial, and let be integers with . Suppose that every irreducible module with is thin and satisfies . If the matrix consisting of the intersection numbers defined by
(2.2) 
(where the entry is ) is nonsingular, then
where .
Proof.
Fix a set of irreducible modules in such that . Observe that
so that by (2.1) we have
In particular, it follows that
and that
for every with .
Pick any with , and let be a nonzero vector in . Recall that , where . First, suppose that . Since is thin and since , for , the vector is nonzero and hence is a basis of . Moreover, for , it follows that
where we have used the fact that is the number of the geodesics between two vertices at distance (in the distanceregular graph ). Since , the vectors are nonzero and hence form a basis of . Thus, since the coefficient matrix (2.2) is nonsingular, the vectors also form a basis of . It follows that . In particular, . Next, suppose that . Likewise, using the fact that the last columns of the matrix (2.2) are linearly independent, we find that the vectors are linearly independent, and hence that . Thus, it follows that
as desired. ∎
Example 2.5.
Example 2.6.
Next, we move on to the polynomial case.
Theorem 2.7.
Suppose that is polynomial, and let be integers with and . If every irreducible module with is dual thin, and satisfies and , then
where .
Proof.
Again, fix a set of irreducible modules in such that . Observe that
so that
In particular, it follows that
and that
for every with .
Pick any with , and let be a nonzero vector in . First, suppose that . Then, . Since is dual thin, is an orthogonal basis of . We note that for . Thus, the vectors belong to and form a basis of , since the coefficient matrix is Vandermonde. It follows that . In particular, . Next, suppose that . Likewise, we find that the vectors belong to and are linearly independent, from which it follows that . Thus, it follows that
where the second equality follows since every with is dual thin and satisfies . This completes the proof. ∎
Example 2.9.
Example 2.10.
Example 2.11.
We note that if some of the assumptions on the irreducible modules in Theorems 2.3 and 2.7 are not satisfied, then the dimensions of the subspaces in question can indeed be smaller. For example, we have the following result:
Proposition 2.12.
Suppose that is polynomial and polynomial, and let . Let be the eigenvalues^{4}^{4}4The are the eigenvalues of the subgraph of induced on (called the local graph), which is regular with valency . of on . For every , let (a Möbius transformation) where , and define . Then, and .
Proof.
Let be as in the proofs of Theorems 2.3 and 2.7. Recall that and act on every as a tridiagonal pair in the sense of [20]; cf. [20, Example 1.4]. In particular, by [20, Lemma 4.5] we have for .
Let , . Let be the second largest and the smallest eigenvalues of , respectively. Then, in view of [18, Lemma 8.5], it follows that the condition that (resp. ) implies that (resp. ). Next, observe that . On the other hand, by [18, Theorem 8.4] we have . Thus, the condition that (resp. ) implies again that (resp. ). With these explained, it follows from [18, Lemma 8.5, Theorems 9.8, 10.1, 11.5] that every is thin, and that