On relative $t$-designs in polynomial association schemes

Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nicely. The two versions of relative $t$-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.


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 R n , the unit sphere S n−1 ⊆ R n , and the set of k-subsets of a given v-set. It has therefore a vast range of applications in various fields of science. See, e.g., [8,2].
The similarities between the theories of spherical t-designs and combinatorial t-(v, k, λ) designs are well known; cf. [14,13,18,1]. Historically, the concept of spherical t-designs was introduced by Delsarte, Goethals, and Seidel [14] as a continuous analogue of that of t-designs in Q-polynomial association schemes due to Delsarte [10,11]. (Combinatorial t-(v, k, λ) designs are precisely the t-designs in the Johnson scheme J(v, k).) It was then generalized to the concept of Euclidean t-designs by Neumaier and Seidel [22], and Euclidean t-designs quickly became an active area of research; cf. [2]. Although the counterparts of Euclidean t-designs in the theory of Q-polynomial association schemes were already defined and discussed to some extent by Delsarte [12] (cf. [3]) much earlier as relative t-designs, it seems that the theory of the latter has not been fully developed yet (except in the case of the binary Hamming scheme H(n, 2), in which case relative t-designs turn out to be equivalent to regular t-wise balanced designs). This paper is a contribution to this theory. Our discussions also include a concept of relative t-designs in general P -polynomial association schemes as well, following Delsarte and Seidel [15].
We refer the reader to [10,5,6,18,21,9], etc., for the background on association schemes and some fundamental concepts. Throughout the paper, let X = (X, {R r } d r=0 ) be a (symmetric) d-class association scheme, and fix a base vertex u 0 ∈ X. Let X r = {x ∈ X | (u 0 , x) ∈ R r } for r = 0, 1, . . . , d. We call X 0 , X 1 , . . . , X d the shells of X. Let F (X) be the vector space consisting of all the real valued functions on X. In the following arguments we often identify F (X) with the vector space R X consisting of the real column vectors with coordinates indexed by X.
We first introduce a concept of t-designs for general P -polynomial association schemes. Suppose that X is P -polynomial with respect to the ordering R 0 , R 1 , . . . , R d . In the study of spherical/Euclidean t-designs in R n , we work with the vector space of polynomials in n variables, in particular with the subspaces of homogeneous polynomials. For the P -polynomial scheme X, it is natural to consider the following subspaces of F (X). For every z ∈ X j , we define f z ∈ F (X) by f z (x) = 1, if x ∈ X i , i ≥ j, and (x, z) ∈ R i−j , 0, otherwise, (x ∈ X).
We now consider a (positive) weighted subset (Y, w) of X, that is to say, a pair of a subset Y of X and a function w : Y → (0, ∞). Let {r 1 , r 2 , . . . , r p } = {r | Y ∩ X r = ∅}, and . . , p. We say that Y is supported by the union S = X r 1 ∪ X r 2 ∪ · · · ∪ X rp of p shells. For any subspace R(X) of F (X), we write R(S) = {f | S | f ∈ R(X)}.
This definition is due to Delsarte and Seidel [15,Section 6] for the binary Hamming scheme H(d, 2). In this paper, we mostly consider the case t = 2e for simplicity. 15]). Let (Y, w) be a relative 2e-design of a Hamming scheme H(d, q) with respect to u 0 in the sense of Definition 1.1. Let S = X r 1 ∪ · · · ∪ X rp be the union of the shells which support Y . Then, |Y | ≥ dim Hom 0 (S) + Hom 1 (S) + · · · + Hom e (S  [15]. Namely, he proved dim Hom 0 (S) + Hom 1 (S) + · · · + Hom e (S) = k e + k e−1 + · · · + k e−p+1 , under a reasonable additional condition which avoids the triviality. In this paper, we focus on generalizing (1.2) to other classes of P -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 t-designs for Q-polynomial association schemes was introduced by Delsarte [12]. We now recall the definition. Suppose that X is Q-polynomial with respect to the ordering E 0 , E 1 , . . . , E d of its primitive idempotents, and let L j (X)(⊆ F (X)) be the column space of E j (j = 0, 1, . . . , d). Then, dim(L j (X)) = rank(E j ) =: m j (j = 0, 1, . . . , d), and we have the following orthogonal direct sum decomposition of F (X): for every f ∈ L 0 (X) ⊥ L 1 (X) ⊥ · · · ⊥ L t (X).
Bannai and Bannai [3] obtained the following Fisher type inequality for general Qpolynomial association schemes: 3]). Let (Y, w) be a relative 2e-design of the Q-polynomial scheme X with respect to u 0 in the sense of Definition 1.3. Let S = X r 1 ∪ · · · ∪ X rp be the union of the shells which support Y . Then, 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 [20] for H(d, 2), but was unsuccessful in general. Then, this attempt lead Xiang to obtain a successful result in the general case for H(d, 2), as it is known that the two definitions of relative t-designs are essentially equivalent for H(d, 2). Namely, both definitions are shown to be equivalent to the geometric definition of relative t-designs coming from the structure of the regular semilattice associated with H(d, 2); cf. [12].  [4]). However, we note that Proposition 1.5. If X is a Hamming scheme H(d, q), then for t = 0, 1, . . . , d, Proof. Without loss of generality, we may suppose that X = {0, 1, . . . , q − 1} d and u 0 = (0, 0, . . . , 0). Let z = (z 1 , z 2 , . . . , z d ) ∈ X j . Note that z has exactly j nonzero entries, and let ℓ 1 , ℓ 2 , . . . , ℓ j be the corresponding coordinates. Then, it is easy to see that f z is the characteristic function of the subset {(x 1 , x 2 , . . . , x d ) ∈ X | x ℓ h = z ℓ h (h = 1, 2, . . . , j)}, which is known to be contained in j i=0 L i (X); see, e.g., [11,25]. 1 Since both sides of (1.4) have the same dimension, we obtain the desired result.
Thus, for H(d, q), relative t-designs in the sense of Definition 1.1 are equivalent to relative t-designs in the sense of Definition 1.3. This observation seems to be new for H(d, q) for general q. As is mentioned before, for H(d, 2), the result of Xiang [34] implies that the right hand side of (1.3) is also given explicitly by dim L 0 (S) + L 1 (S) + · · · + L e (S) = m e + m e−1 + · · · + m e−p+1 , (1.5) since m j = k j (j = 0, 1, . . . , d) in this case. In a private communication, Xiang extended his main result in [34] to general q. Thus, the right hand side of (1.3) is also given explicitly as (1.5) for H(d, q). 1 In Appendix B, we give a direct proof that f z belongs to j i=0 L i (X), which does not use the theory of regular semilattices found in [11,25].
In this paper, we investigate to what extent the above results can be generalized to other P -and/or Q-polynomial association schemes. In Section 2, we derive sufficient conditions that (1.2) (resp. (1.5)) holds for a P -polynomial (resp. Q-polynomial) association scheme (Theorems 2.3 and 2.7). These conditions can be readily checked for H(d, q), 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) self-dual P -and Q-polynomial association schemes, but it turns out that this is not the case in general. Indeed, in Section 3, we show that if X is formally self-dual, P -polynomial (and thus Q-polynomial), and satisfies Hom 0 (X) + Hom 1 (X) = L 0 (X) + L 1 (X), then X must be a Hamming scheme H(d, q), provided that d ≥ 6 (Theorem 3.2). All of these theorems are proved using the theory of the Terwilliger algebra [29,30,31]. See [27] for more applications of the Terwilliger algebra to design theory.

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 C X instead of R X , 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 x ∈ X, letx ∈ F (X) = C X be the characteristic function of the set {x}. Let A 0 , A 1 , . . . , A d and E 0 , E 1 , . . . , E d be (fixed orderings of) the adjacency matrices and the primitive idempotents of X, respectively.
They form two bases of the dual Bose-Mesner algebra with respect to u 0 . When we assume that X is Ppolynomial (resp. Q-polynomial), we understand that A 0 , A 1 , . . . , A d (resp. E 0 , E 1 , . . . , E d ) is the P -polynomial ordering (resp. Q-polynomial ordering) and write The Terwilliger algebra T with respect to u 0 is the subalgebra of the full matrix algebra generated by the Bose-Mesner algebra and the dual Bose-Mesner algebra. We note that T is semisimple since it is closed under conjugatetransposition.
The endpoint, dual endpoint, diameter, and the dual diameter of an irreducible T - Lemma 3.6]. It is thin, dual thin, and has diameter and dual diameter both equal to d. We call X thin (resp. dual thin) with respect to u 0 if every irreducible T -module is thin (resp. dual thin). 3 The next two lemmas will be freely used in our discussions. Lemma 3.9]). Suppose that X is P -polynomial. Let W be an irreducible T -module and set ρ = ρ(W ), δ = δ(W ). Then, the following hold: . Then, the following hold: We note that if X is P -polynomial then for j = 0, 1, . . . , d.
Suppose that X is P -polynomial, and let e, r 1 , r 2 , . . . , r p be integers with so that by (2.1) we have In particular, it follows that Pick any W ∈ W with ρ := ρ(W ) ≤ e, and let v be a nonzero vector in E * ρ W . Recall that {i | E * i W = 0} = {ρ, . . . , ρ + δ}, where δ = δ(W ). First, suppose that ρ ≤ e − p + 1. Since W is thin and since ρ + δ ≥ e, for j = e − p + 1, . . . , e, the vector v j = E * j A j−ρ v is nonzero and hence is a basis of E * j W . Moreover, for j = e − p + 1, . . . , e, it follows that where we have used the fact that c n . . . c 2 c 1 is the number of the geodesics between two vertices at distance n (in the distance-regular graph (X, R 1 )). Since r 1 , r 2 , . . . , r p ∈ {ρ, . . . , ρ + δ}, the vectors E * r i A r i −e+p−1 v e−p+1 (i = 1, 2, . . . , p) are nonzero and hence form a basis of p i=1 E * r i W . Thus, since the coefficient matrix (2.2) is nonsingular, the vectors In particular, dim e j=0 Hom j (S) ∩ W = p. Next, suppose that e − p + 2 ≤ ρ ≤ e. Likewise, using the fact that the last (e − ρ + 1) columns of the matrix (2.2) are linearly independent, we find that the vectors . . , e) are linearly independent, and hence that dim  Example 2.6. Suppose that X is a dual polar scheme. Then, c i = (q i − 1)/(q − 1) (i = 1, 2, . . . , d) for some prime power q ≥ 2. Thus, the matrix (2.2) is again essentially Vandermonde (in the variables q r 1 , q r 2 , . . . , q rp ), and hence is nonsingular. We note that X is thin; cf. [31, Example 6.1]. In Appendix A, we show that Theorem 1.2 is valid for dual polar schemes.
Next, we move on to the Q-polynomial case.

Proof. Again, fix a set W of irreducible T -modules in
In particular, it follows that Pick any W ∈ W with ρ * := ρ * (W ) ≤ e, and let v be a nonzero vector in E ρ * W . First, suppose that ρ * ≤ e − p + 1.
is an orthogonal basis of W . We note that E * r i W = 0 for i = 1, 2, . . . , p. Thus, the vectors In particular, dim e j=0 L j (S) ∩ W = p. Next, suppose that e − p + 2 ≤ ρ * ≤ e. Likewise, we find that the vectors  H(d, q). Then, ρ(W ) = ρ * (W ) for every irreducible T -module W ; cf. [31, Example 6.1]. Thus, the assumption of Theorem 2.7 is satisfied provided that e ≤ r 1 < r 2 < · · · < r p ≤ d − e. Of course, in this case the conclusion also follows from Proposition 1.5, Theorem 2.3, and Example 2.5.
Proof. Since X is formally self-dual, in the notation of [5,Section 3.5] and [29,Section 2], the parameters of X satisfy one of the following cases and this is independent of i by Proposition 3.1, so that for i = 1, 2, . . . , d − 1, and this identity is valid for i = d as well. However, as polynomials in q i , the left hand side is of degree five, whereas the right hand side is of degree four. Since d ≥ 6, this is impossible. Case (I) with s = s * = 0 is ruled out in the same way. Next, consider Case (II) with s = s * . Then, it follows that Again, as polynomials in i, the denominator must be a scalar multiple of the numerator.
If X satisfies Case (III) with s = s * , then by the classification due to Terwilliger [28], it follows that X is isomorphic to H(d, 2) (d even) or the bipartite half of H(2d + 1, 2), but with respect to the second P -polynomial orderings. 7 We have c i = i (i = 1, 2, . . . , d) in either case, and it follows that c i /(θ * i − θ * 0 ) cannot be constant, since θ * 0 , θ * 1 , . . . , θ * d are not an arithmetic progression.
Thus, we are left with Case (IIC). In this case, by the classification due to Egawa [16], X is a Hamming scheme or a Doob scheme. If X is a Hamming scheme, then we are done. Thus, suppose that X is a Doob scheme. Then, there is a thin irreducible T -module W with ρ(W ) = 1, ρ * (W ) = 2, and δ(W ) = d − 2. This fact follows from Tanabe's description [26] of the irreducible T -modules of Doob schemes, but we may also prove it as follows. The local graph of the Doob graph (X, R 1 ) (whose adjacency matrix is essentially E * 1 AE * 1 ) is a disjoint union of hexagons and 3-cliques, so that it has −2 as an eigenvalue. On the other hand, we have Thus, by [17,Theorem 9.8], any eigenvector (in E * 1 C X ) of E * 1 AE * 1 with eigenvalue −2 generates such a T -module. Now, let v be a nonzero vector in E * 1 W . Then, v is nonzero and belongs to Hom 1 (X). However, since ρ * (W ) = 2, it is contained in L 2 (X) + L 3 (X) + · · · + L d (X). Thus, we conclude that Hom 0 (X) + Hom 1 (X) = L 0 (X) + L 1 (X), and the proof is complete.

A Comments on Theorem 1.2
In this appendix, we generalize Theorem 1.2 to dual polar schemes (Theorem A.6). Suppose that X is a dual polar scheme, so that X is the set of maximal isotropic subspaces of a vector space V over a finite field, equipped with a non-degenerate form (alternating, Hermitian, or quadratic) of Witt index d. For convenience, we shall work with the dual polar graph (X, R 1 ) with path-length distance ∂.
For the moment, fix x, y ∈ X and write i = ∂(u 0 , x), j = ∂(u 0 , y), h = ∂(x, y), and ℓ = dim(u 0 ∩ U), where U = x ∩ y. We note that ℓ ≥ d − i − j. Our goal is to show that f x f y ∈ Hom d−ℓ (X). We set X ′ = {z ∈ X | U ⊆ z}, and observe that X ′ induces a dual polar graph with diameter h.
Lemma A.4. For every z ∈ X such that f x (z) = f y (z) = 1, there is a unique z ′ ∈ X ′ such that f x (z ′ ) = f y (z ′ ) = 1 and f z ′ (z) = 1.
Proposition A.5. With the above notation, it holds that f z ∈ Hom d−ℓ (X).

Proof. Immediate from Lemmas A.3 and A.4.
Theorem A.6. Theorem 1.2 is valid for dual polar schemes.
B Comments on Proposition 1.5 We use the notation in the proof of Proposition 1.5. We mentioned there that the function f z belongs to j i=0 L i (X). While this fact is just a special case of a more general result about regular semilattices [11,12,25], we now provide an independent proof.
Assume that i > j, and pick any y = (y 1 , y 2 , . . . , y d ) ∈ X i . Then, the (standard) Hermitian inner product between ε y and f z is given by j h=1 ζ z ℓ h y ℓ h   ℓ =ℓ 1 ,...,ℓ j q−1 Since i > j, there is an ℓ = ℓ 1 , . . . , ℓ j such that y ℓ = 0. For this ℓ, we have q−1 x ℓ =0 ζ x ℓ y ℓ = 0. Thus, f z is orthogonal to ε y . It follows that f z is orthogonal to d i=j+1 L i (X), and hence it is contained in j i=0 L i (X), as desired.