Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.


Introduction
This paper is a contribution to the study of relative t-designs in Q-polynomial association schemes.In the Delsarte theory [16], the concept of t-designs is introduced for arbitrary Q-polynomial association schemes.For the Johnson scheme J(n, k), the t-designs in the sense of Delsarte are shown to be the same thing as the combinatorial t-(n, k, λ) designs.There are similar interpretations of t-designs in some other important families of Q-polynomial association schemes [16,17,19,34,41].The concept of relative t-designs is also due to Delsarte [18], and is a relaxation of that of t-designs.Relative t-designs can again be interpreted in several cases, including J(n, k).For the n-dimensional hypercube Q n (or the binary Hamming scheme H(n, 2)) which will be our central focus in this paper, these are equivalent to the weighted regular t-wise balanced designs, which generalize the combinatorial t-(n, k, λ) designs by allowing multiple block sizes as well as weights.
The Delsarte theory has a counterpart for the unit sphere S n−1 in R n , established by Delsarte, Goethals, and Seidel [20].The t-designs in S n−1 are commonly called the spherical t-designs, and are essentially the equally-weighted cubature formulas of degree t for the spherical integration, a concept studied extensively in numerical analysis.Spherical t-designs were later generalized to Euclidean t-designs by Neumaier and Seidel [35] (cf.[21]).Euclidean t-designs are in general supported on multiple concentric spheres in R n , and it follows that we may think of them as the natural counterpart of relative t-designs in R n .This point of view was discussed in detail by Bannai and Bannai [3].See also [7,8].The success and the depth of the theory of Euclidean t-designs (cf.[38]) has been one driving force for the recent research activity on relative t-designs in Q-polynomial association schemes; see, e.g., [3,5,6,7,8,9,11,32,51,53,54].
A relative t-design in a Q-polynomial association scheme (X, R) is often defined as a certain weighted subset of the vertex set X, i.e., a pair (Y, ω) of a subset Y of X and a function ω : Y → (0, ∞).We are given in advance a 'base vertex' x ∈ X, and (Y, ω) gives a 'degree-t approximation' of the shells (or spheres or subconstituents) with respect to x on which Y is supported.See Sections 2 and 3 for formal definitions.Bannai and Bannai [3] proved a Fisher-type lower bound on |Y |, and we call (Y, ω) tight if it attains this bound.We may remark that t must be even in this case.In this paper, we continue the study (cf.[5,9,32,51,53]) of tight relative t-designs in the hypercubes Q n , which are one of the most important families of Q-polynomial association schemes.The Delsarte theory directly applies to the tight relative t-designs in Q n supported on one shell, say, the k th shell, as these are equivalent to the tight combinatorial t-(n, k, λ) designs.(We note that the k th shell induces J(n, k).)Our aim is to extend this structure theory to those supported on two shells.We may view the results of this paper roughly as counterparts to (part of) the results by Bannai and Bannai [2,4] on tight Euclidean t-designs on two concentric spheres.
Let t = 2e be even.In Theorem 5.3, which is our first main result, we show under a mild condition that a tight relative 2e-design in Q n supported on two shells induces the structure of a coherent configuration with two fibers.Moreover, from this structure we deduce that a certain polynomial of degree e, known as a Hahn polynomial, must have only integral simple zeros.We note that the case e = 1 was handled previously by Bannai, Bannai, and Bannai [5].The Hahn polynomials are a family of hypergeometric orthogonal polynomials in the Askey scheme [31,Section 1.5], and that their zeros are integral provides quite a strong necessary condition on the existence of such relative 2e-designs.The corresponding necessary condition for the tight combinatorial 2e-(n, k, λ) designs from the Delsarte theory was used successfully by Bannai [1]; that is to say, he showed that, for each given integer e 5, there exist only finitely many non-trivial tight 2e-(n, k, λ) designs, where n and k (and thus λ) vary.See also [22,36,52].We extend Bannai's method to prove our second main result, Theorem 7.1, which presents a version of his theorem for our case.
The sections other than Sections 5 and 7 are organized as follows.We collect the necessary background material in Sections 2 and 3. Section 3 also includes a few general results on relative t-designs in Q-polynomial association schemes.As in [6,44], our main tool in the analysis of relative t-designs is the Terwilliger algebra [46,47,48], which is a non-commutative semisimple C-algebra containing the adjacency algebra.Section 4 is devoted to detailed descriptions of the Terwilliger algebra of Q n .It is well known (cf.[30,31]) that the Hahn polynomials ( 3 F 2 ) degenerate to the Hermite polynomials ( 2 F 0 ) by an appropriate limit process, and a key in Bannai's method above was to evaluate precisely the behavior of the zeros of the Hahn polynomials in this process.In Section 6, we revisit this part of the method in a form suited to our purpose.Our account will also be simpler than that in [1].In Appendix, we provide a proof of a number-theoretic result (Proposition 7.2) which is a variation of a result of Schur [40,Satz I].

Coherent configurations and association schemes
We begin by recalling the concept of coherent configurations.
Definition 2.1.The pair (X, R) of a finite set X and a set R of non-empty subsets of X 2 is called a coherent configuration on X if it satisfies the following (C1)-(C4): Remark 2.2.Suppose that a finite group G acts on X, and let R be the set of the orbitals of G, that is to say, the orbits of G in its natural action on X 2 .Then (X, R) is a coherent configuration.Moreover, (X, R) is homogeneous (resp.an association scheme) if and only if the action of G on X is transitive (resp.generously transitive, i.e., for any x, y ∈ X we have (x g , y g ) = (y, x) for some g ∈ G).
Let (X, R) be a coherent configuration as above.For every R ∈ R 0 , let Φ R be the subset of X such that R = {(x, x) : x ∈ Φ R }.Then we have We call the Φ R (R ∈ R 0 ) the fibers of (X, R).By setting in (C4) either R ∈ R 0 and S = T , or S ∈ R 0 and R = T , it follows that for every T ∈ R, we have T ⊂ Φ R × Φ S for some R, S ∈ R 0 .In particular, (X, R) is homogeneous whenever it is an association scheme.Let , which is symmetric by (C3), is called the type of (X, R).
Let M X (C) be the C-algebra of all complex matrices with rows and columns indexed by X, and let V = C X be the C-vector space of complex column vectors with coordinates indexed by X.We endow V with the Hermitian inner product where † denotes adjoint.For every R ∈ R, let A R ∈ M X (C) be the adjacency matrix of the graph (X, R) (directed, in general), i.e., Then (C1)-(C4) above are rephrased as follows: (A1)

R∈R
A R = J (the all-ones matrix).

R∈R0
A R = I (the identity matrix).
Let A = span{A R : R ∈ R}.Then from (A2) and (A4) it follows that A is a subalgebra of M X (C), called the adjacency algebra of (X, R).We note that A is semisimple as it is closed under † by virtue of (A3).By (A1), A is also closed under entrywise (or Hadamard or Schur ) multiplication, which we denote by •.The A R are the (central) primitive idempotents of A with respect to •, i.e., Remark 2.3.If (X, R) arises from a group action as in Remark 2.2, then A coincides with the centralizer algebra (or Hecke algebra or commutant) for the corresponding permutation representation g → P g (g ∈ G) on V , i.e., A subalgebra of M X (C) is called a coherent algebra if it contains J, and is closed under • and † .We note that the coherent algebras are precisely the adjacency algebras of coherent configurations.It is clear that the intersection of coherent algebras in M X (C) is again a coherent algebra.In particular, for any subset S of M X (C), we can speak of the smallest coherent algebra containing S, which we call the coherent closure of S.
From now on, we assume that (X, R) is an association scheme.As is the case for many examples of association schemes, we write and say that (X, R) has n classes.We will then abbreviate p k i,j = p R k Ri,Rj , A i = A Ri , and so on.The adjacency algebra A is commutative in this case, and hence it has a basis E 0 , E 1 , . . ., E n consisting of the (central) primitive idempotents, i.e., Put differently, E 0 V, E 1 V, . . ., E n V are the maximal common eigenspaces (or homogeneous components or isotypic components) of A, and the E i are the corresponding orthogonal projections.Since the A i are real symmetric matrices, so are the E i .Note that the matrix |X| −1 J ∈ A is an idempotent with rank one, and thus primitive.We will always set For convenience, we let Though our focus in this paper will be on Q-polynomial association schemes, we first recall the P -polynomial property for completeness.We say that the association scheme (X, R) is P -polynomial (or metric) with respect to the ordering A 0 , A 1 , . . ., A n if there are non-negative integers where b −1 and c n+1 are indeterminates.In this case, A 1 recursively generates A, and hence has n + 1 distinct eigenvalues θ 0 , θ 1 , . . ., θ n ∈ R, where we write (1) We note that (X, R) is P -polynomial as above precisely when the graph (X, R 1 ) is a distance-regular graph and (X, R i ) is the distance-i graph of (X, R 1 ) (0 i n).
See, e.g., [10,12,15,27] for more information on distance-regular graphs.We say that (X, R) is Q-polynomial (or cometric) with respect to the ordering where b * −1 and c * n+1 are indeterminates.In this case, |X|E 1 recursively generates A with respect to •, and hence has n + 1 distinct entries θ * 0 , θ * 1 , . . ., θ * n ∈ R, where we write We call the θ * i the dual eigenvalues of |X|E 1 .We may remark that E 1 • E i , being a principal submatrix of E 1 ⊗E i , is positive semidefinite, so that the scalars a * i , b * i , and c * i are non-negative (the so-called Krein condition).The Q-polynomial association schemes are an important subject in their own right, and we refer the reader to [23,29] and the references therein for recent activity.Below we give two fundamental examples of P -and Q-polynomial association schemes, both of which come from transitive group actions.See [10,12,16] for the details.
Example 2.4.Let v and k be positive integers with v > k, and let X be the set of k-subsets of {1, 2, . . ., v}.Set n = min{k, v − k}.For x, y ∈ X and 0 i n, we let (x, y) ∈ R i if |x ∩ y| = k − i.The R i are the orbitals of the symmetric group S v acting on X.We call (X, R) a Johnson scheme and denote it by J(v, k).The eigenvalues of A 1 are given in decreasing order by and J(v, k) is Q-polynomial with respect to the corresponding ordering of the E i (cf.(1)).
Example 2.5.Let q 2 be an integer and let X = {0, 1, . . ., q − 1} n .For x, y ∈ X and 0 i n, we let (x, y) ∈ R i if x and y differ in exactly i coordinate positions.The R i are the orbitals of the wreath product S q ≀ S n of the symmetric groups S q and S n acting on X.We call (X, R) a Hamming scheme and denote it by H(n, q).The eigenvalues of A 1 are given in decreasing order by and H(n, q) is Q-polynomial with respect to the corresponding ordering of the E i (cf.( 1)).The Hamming scheme H(n, 2) is also known as the n-cube (or ndimensional hypercube) and is denoted by Q n .
Assumption 2.6.For the rest of this section and in Section 3, we assume that (X, R) is an association scheme and is Q-polynomial with respect to the ordering E 0 , E 1 , . . ., E n of the primitive idempotents.
In general, for any positive semidefinite Hermitian matrices B, C ∈ M X (C), we have (cf.[45]) where Hence it follows from (2) that from which it follows that for 0 h, k n.See also [10,Section 2.8].
We now fix a 'base vertex' x ∈ X.Let We call the X i the shells (or spheres or subconstituents) of (X, R) with respect to x.For every i (0 i n), define the diagonal matrix Then we have We call the E * i the dual idempotents of (X, R) with respect to x.The subspace A * = A * (x) = span{E * 0 , E * 1 , . . ., E * n } is then a subalgebra of M X (C), which we call the dual adjacency algebra of (X, R) with respect to x.The Terwilliger algebra (or subconstituent algebra) of (X, R) with respect to x is the subalgebra T = T (x) of M X (C) generated by A and A * [46,47,48].We note that T is semisimple as it is closed under † .Remark 2.7.If (X, R) arises from a group action as in Remark 2.2, which we recall is generously transitive in this case, then T is a subalgebra of the centralizer algebra for the action of the stabilizer G x of x in G.The two algebras are known to be equal, e.g., for J(v, k) and H(n, q); see [25,43].
For every subset Y of X, let Ŷ ∈ V be the characteristic vector of Y , i.e., ( Ŷ ) y = 1 if y ∈ Y, 0 otherwise, (y ∈ X).
In particular, X denotes the all-ones vector in V .We will simply write x for the characteristic vector of the singleton {x}.With this notation established, we have from which it follows that (6) T x = span{ Xi : 0 i n} = span{E i x : 0 i n}.
The T -module T x is easily seen to be irreducible with dimension n + 1 (cf.[46,Lemma 3.6]), and is called the primary T -module.We define the dual adjacency matrix Since the θ * i are mutually distinct, A * 1 generates A * .Moreover, since it follows from (4) that (8) Let W be an irreducible T -module.We define the dual support W * s , the dual endpoint r * (W ), and the dual diameter d * (W ) of W by We note that the primary T -module T x is dual thin, and that it is a unique irreducible T -module up to isomorphism which has dual endpoint zero or dual diameter n.The following lemma is an easy consequence of (8): Lemma 3.12]).With reference to Assumption 2.6, write Let W be an irreducible T -module and set r * = r * (W ), d * = d * (W ).Then the following hold: In particular, W = A * E r * W .

Relative t-designs in Q-polynomial association schemes
In this section, we develop some general theory on relative t-designs in Qpolynomial association schemes.
Recall Assumption 2.6.Throughout this section, we fix a base vertex x ∈ X, and write , and T = T (x).In Introduction, we meant by a weighted subset of X a pair (Y, ω) of a subset Y of X and a function ω : Y → (0, ∞).For convenience, however, we extend the domain of ω to X by setting ω(y) = 0 for every y ∈ X\Y .We will also naturally identify V with the set of complex functions on X, so that ω ∈ V and Y = supp ω.In our discussions on relative t-designs, we will often consider the set ( 9) and say that (Y, ω) is supported on ℓ∈L X ℓ .
For comparison, we begin with the algebraic definition of t-designs in (X, R) due to Delsarte [16,17].
Delsarte [18] generalized this concept as follows: Remark 3.3.Delsarte introduced the concept of t-designs for subsets Y of X in [16], i.e., when ω = Ŷ , whereas in [17,18] he mostly considered general (i.e., not necessarily non-negative) non-zero vectors ω ∈ V in the discussions on t-designs and relative t-designs.Some facts/results below, such as Examples 3.4 and 3.5, Proposition 3.6, and Theorem 3.8, are still valid for general ω ∈ V , but the Fishertype lower bound on |Y | = | supp ω| (cf.Theorem 3.9) makes sense only when ω is non-negative.
For the Johnson and Hamming schemes, Delsarte [16,17,18] showed that these algebraic concepts indeed have geometric interpretations: Example 3.4.Let (X, R) be the Johnson scheme J(v, k) from Example 2.4.Then (Y, ω) is a t-design if and only if, for every t-subset z of {1, 2, . . ., v}, the sum λ z of the values ω(y) over those y ∈ Y such that z ⊂ y, is a constant independent of z.On the other hand, (Y, ω) is a relative t-design if and only if the above λ z depends only on |x ∩ z|.We note that (Y, Ŷ ) is a t-design if and only if Y is a t-(v, k, λ) design (cf.[13,Chapter II.4]) for some λ.
Example 3.5.Let (X, R) be the Hamming scheme H(n, q) from Example 2.5.Then (Y, ω) is a t-design if and only if, for every t-subset T of {1, 2, . . ., n} and every function f : T → {0, 1, . . ., q − 1}, the sum λ T ,f of the values ω(y) over those y = (y 1 , y 2 , . . ., y n ) ∈ Y such that y i = f (i) (i ∈ T ), is a constant independent of the pair (T , f ).On the other hand, (Y, ω) is a relative t-design if and only if the above λ T ,f depends only on |{i ∈ T : x i = f (i)}|, where x = (x 1 , x 2 , . . ., x n ).We note that (Y, Ŷ ) is a t-design if and only if the transpose of the |Y |×n matrix formed by arranging the elements of Y (in any order) is an orthogonal array OA(|Y |, n, q, t) (cf.[13,Chapter III.6]).For the case q = 2, i.e., for Q n , if we choose the base vertex as x = (0, 0, . . ., 0), then (Y, Ŷ ) is a relative t-design if and only if Y is a regular t-wise balanced design of type t-(n, L, λ) (cf.[38,Section 4.4]) for some λ, where L is from (9), and where we identify the elements of X = {0, 1} n with their supports.
Similar results hold for some other important families of P -and Q-polynomial association schemes; see, e.g., [17,18,19,34,41].Proposition 3.6 (cf.[3,Theorem 4.5]).With reference to Assumption 2.6, let (Y, ω) be a weighted subset supported on ℓ∈L X ℓ .Then we have (10) ω| where ω| T x denotes the orthogonal projection of ω on the primary T -module T x.Moreover, (Y, ω) is a relative t-design if and only if (6).The first part follows since the Xi form an orthogonal basis of The second part is also immediate from Hence, if (Y, ω) is a relative t-design such that X ℓ ⊂ Y for some ℓ, and if ω is constant on X ℓ , then the weighted subset (Y \X ℓ , (I −E * ℓ )ω) obtained by discarding X ℓ from Y is again a relative t-design.This observation is particularly important when applying Theorem 3.8 below; for example, we can always assume that 0 ∈ L.
The following is a slight generalization of Delsarte's Assmus-Mattson theorem for Q-polynomial association schemes [18,Theorem 8.4], and can also be viewed as a variation of [9,Theorem 3.3], which in turn generalizes [28,Proposition 1].See also [11].The proof is in fact identical to that of [44,Theorem 4.3], but we include it below because of the potential importance of the result.Theorem 3.8.With reference to Assumption 2.6, let (Y, ω) be a relative t-design Proof.Let U = (T x) ⊥ be the orthogonal complement of T x in V , which we recall is the sum of all the non-primary irreducible T -modules in V .On the one hand, we have Since A * 1 generates A * and has at most |L| distinct eigenvalues on this subspace (cf.(7)), it follows that (11) Hence it follows from ( 8) and ( 11) that In particular, for every ℓ ∈ L we have In other words, (Y ∩ X ℓ , E * ℓ ω) is a relative (t − |L| + 1)-design, as desired.
Bannai and Bannai [3, Theorem 4.8] established the following Fisher-type lower bound on the size of a relative t-design with t even: Theorem 3.9.With reference to Assumption 2.6, let (Y, ω) be a relative 2e-design (e ∈ N) supported on ℓ∈L X ℓ .Then Recall from Example 3.5 that the relative t-designs in the hypercubes are equivalent to the weighted regular t-wise balanced designs.
Example 3.11.Let (X, R) be the n-cube Q n from Example 2.5.Xiang [51] showed that if e ℓ n − e for every ℓ ∈ L, then (12) dim We may remark that (cf.[12, Theorem 9.2.1]) See also [32]  Let (Y, ω) be a tight relative 2e-design supported on ℓ∈L X ℓ .Bannai, Bannai, and Bannai [5, Theorem 2.1] showed that if the stabilizer of x in the automorphism group of (X, R) acts transitively on each of the shells X i then ω is constant on Y ∩ X ℓ for every ℓ ∈ L. The next theorem generalizes this result by replacing group actions by combinatorial regularity.Observe that the fibers of the coherent closure of T are in general finer than the shells X i .Theorem 3.14.With reference to Assumption 2.6, let (Y, ω) be a tight relative 2e-design (e ∈ N) supported on ℓ∈L X ℓ .For every ℓ ∈ L, the weight ω is constant on Y ∩ X ℓ provided that X ℓ remains a fiber of the coherent closure of T .
Proof.Let (cf.( 10)) Note that D ∈ T .Let F be the orthogonal projection onto BV , where Observe that BV = (BB † )V, and that F is written as a polynomial in the Hermitian (in fact, real and symmetric) matrix BB † .In particular, F ∈ T .
Since (Y, ω) is tight, we have Let u 1 , u 2 , . . ., u |Y | be an orthonormal basis of BV , and let Then we have where | Y ×Y etc. mean taking corresponding submatrices.Note that these are square matrices, and that D ′ and D′ are invertible.Then it follows that Indeed, since we may write it follows from (5) (applied to h = k = e) and Proposition 3.6 that the (i, j)-entry of the LHS in ( 15) is equal to where means complex conjugate.By ( 14) and ( 15), we have In particular, F ′ is a diagonal matrix.Now, let ℓ ∈ L and suppose that X ℓ remains a fiber of the coherent closure of T .Then the (y, y)-entry of F ∈ T is constant for y ∈ X ℓ (cf.(A1) and (A2)), and the same is true for D. Hence it follows from ( 16) that ω(y) = D y,y must be constant for y ∈ Y ∩ X ℓ .This completes the proof.

The Terwilliger algebra of Q n
For the rest of this paper, we will focus on relative t-designs in the n-cube Q n from Example 2.5.We will need detailed descriptions of the Terwilliger algebra of Q n and its irreducible modules, and we collect these in this section.Thus, we assume that (X, R) = Q n , where X = {0, 1} n .We again fix a base vertex x ∈ X, and write , and T = T (x).The Q-polynomial ordering we consider is the one given in Example 2.5.2Proposition 4.1 (cf.[39, Section I.C]).We have (17) T = span{E * i A j E * k : 0 i, j, k n}.In particular, T is a coherent algebra.
Proof.The RHS in ( 17) is a subspace of T .Recall from Example 2.5 that Q n admits the action of G = S 2 ≀ S n .The stabilizer G x of x in G is isomorphic to S n , and it is immediate to see that every orbital of G x is of the form for some i, j, and k, where the corresponding adjacency matrix is E * i A j E * k .Hence the RHS in (17) agrees with the centralizer algebra for the action of G x on X, which is a coherent algebra; cf.Remark 2.3.Since T is generated by the A i and the E * i , the result follows.Lemma 4.2.For 0 i, j, k n, we have Proof.Routine.
Next we recall basic facts about the irreducible T -modules.Let W be an irreducible T -module.We define the support W s , the endpoint r(W ), and the diameter d(W ) of W by There is a surjective homomorphism U (sl 2 (C)) → T such that (cf.[26, Lemma 7.5]) Every irreducible T -module is then irreducible as an sl 2 (C)-module.We also obtain another surjective homomorphism U (sl 2 (C)) → T by interchanging A 1 and A * 1 and replacing the E i by the E * i above; cf.[26,Lemma 5.3].From now on, we fix an orthogonal irreducible decomposition and fix a unit vector by Theorem 4.3, it follows from (13) that We note that For W, W ′ ∈ Λ r , we observe that the linear map W → W ′ defined by Then we have and from ( 22) and ( 23) it follows that Ȇi,j for 0 r, r ′ ⌊n/2⌋, r i, j n − r, and r ′ i ′ , j ′ n − r ′ .By Theorem 4.3 and Wedderburn's theorem (cf.[14, Section 3]), T is isomorphic to the direct sum of full matrix algebras and the Ȇi,j r form an orthogonal basis of T .See also [24,Section 2].We note that (26) We now recall the Hahn polynomials [31, Section 1.5] where Our aim is to describe the entries of the Ȇi,j r .In view of (25), we will assume for the rest of this section that 0 i j n.
In either case, it follows that for 0 i j n and 0 r min{i, n − j}.

Tight relative 2e-designs on two shells in Q n
We retain the notation of the previous sections.In this section, we discuss tight relative 2e-designs (Y, ω) in Q n supported on two shells X ℓ ⊔ X m , i.e., L = {ℓ, m} (cf.( 9)).Recall from ( 12) that we have in this case but recall also that this is valid under the additional condition that e ℓ, m n−e.
However, both (Y ∩ X ℓ , E * ℓ ω) and (Y ∩ X m , E * m ω) are relative (2e − 1)-designs by Theorem 3.8, so that if ℓ < 2e or ℓ > n − 2e for example, then (Y ∩ X ℓ , E * ℓ ω) must be trivial in view of Example 3.5, i.e., X ℓ ⊂ Y and ω is constant on X ℓ , and hence (Y ∩ X m , E * m ω) is by itself a relative 2e-design; cf.Remark 3.7.This shows that the above condition is not a restrictive one.We also note that Lemma 5.1.Let (Y, ω) be a relative t-design in Q n supported on ℓ∈L X ℓ .Then (Y ′ , A n ω) is a relative t-design supported on ℓ∈L X n−ℓ , where Y ′ = {y ′ : y ∈ Y }, and for every y ∈ X, y ′ denotes the unique vertex such that (y, y ′ ) ∈ R n .

Proof. Immediate from
In view of the above comments, we now make the following assumption: Assumption 5.2.In this section, let (Y, ω) be a tight relative 2e-design (e ∈ N) in Q n supported on two shells X ℓ ⊔ X m , where e ℓ < m n − ℓ ( n − e).
Our aim is to show that Y then induces the structure of a coherent configuration with two fibers, and to obtain a necessary condition on the existence of such (Y, ω) akin to Delsarte's theorem on tight 2e-designs.To this end, we first recall the proof of (12) given in [6, Theorem 2.7, Example 2.9] under the above assumption.

2, and hence dim E *
L E e W = 1 in this case.Suppose next that 0 r < e.On the one hand, since On the other hand, it follows from ( 8) that and hence Moreover, we have (cf.( 7)) Note that in this case we in fact have Combining these comments, we now obtain ( 12) as follows: where we have used (13) and (20).By the above discussions, the set of vectors below forms an orthogonal basis of the subspace (31): As in the proof of Theorem 3.14, let D = diag ω.
We next apply √ D to the above basis vectors and compute their inner products.
where means complex conjugate, and Observe that u belongs to 2e i=0 E i V by (5) (applied to h = k = e) and (32).Hence, by Proposition 3.6 we have Likewise, we have Next, let W ∈ e−1 r=0 Λ r and W ′ ∈ Λ e .Then, by the same argument we have Finally, let W, W ′ ∈ Λ e .In this case, we have Since (Y, ω) is a tight relative 2e-design, it follows from ( 34)-( 38) that the set of vectors below is an orthogonal basis of the subspace For convenience, set We will naturally make the following identification by discarding irrelevant entries: We then define a characteristic matrix H of (Y, ω) by We note that H is a square matrix of size |Y | = n e + n e−1 .By ( 22), (23), and ( 34)- (38), and since we have where Let K denote the diagonal matrix on the RHS in (39).Then it follows that , where we write (41) κ ℓ e = κ m e := κ e for brevity.In particular, we have (42) 1 Moreover, from (39) and ( 40) it follows that r , H m r are non-zero.Likewise, by setting for brevity, we have ) are non-zero, it follows from ( 47)-( 50) that the matrices (κ r ) −1 H ℓ r (H m r ) † (0 r < e) are non-zero and are linearly independent.
It follows from Theorem 3.14 and Proposition 4.1 that ω is constant on each of Y ℓ and Y m , from which it follows that (51) Hence, by comparing with the formula (24) for the matrices Ȇi,j r , we have where a ℓ,ℓ r etc. are in C, and we are again using the notation (41).By ( 43)-( 50) and the above comments, H ′ is a C-algebra with (58) dim where b ℓ,ℓ j etc. are in C. Then H is a C-vector space with (60) dim Note that H is closed under •.By ( 52)-( 56) and Proposition 4.1 (or ( 30)), H ′ is a subspace of H.By ( 30), ( 44), (52), and ( 53), we have Hence it follows that {ξ = 0 : 2ξ ∈ S ℓ,ℓ (Y )} is a set of zeros of the polynomial Note that ψ ℓ,ℓ e (ξ) has degree exactly e, from which it follows that (63) Likewise, we find that {ξ = 0 : 2ξ ∈ S m,m (Y )} is a set of zeros of the polynomial and hence that Finally, by (30), (42), and (57), we have Hence it follows that {ξ and that By (60), (63), (65), and (67), we have Since H ′ is a subspace of H, it follows from (58) that H = H ′ .In particular, H is a C-algebra.It is also clear that H is closed under † and contains J |Y | .We now conclude that H is a coherent algebra.Note also that equality holds in each of (63), (65), and (67).To summarize: Theorem 5.3.Recall Assumption 5.2.With the above notation, the following hold: (i) The set H from (59) is a coherent algebra of type e+1 e e e+1 .(ii) The sets of zeros of the polynomials ψ ℓ,ℓ e (ξ), ψ m,m e (ξ), and ψ ℓ,m e (ξ) from (62), (64), and (66) are given respectively by In particular, the zeros of these polynomials are integral.
Concerning the scalars ω ℓ and ω m appearing in the polynomials ψ ℓ,ℓ e (ξ) and ψ m,m e (ξ), it follows that Proposition 5.4.Recall Assumption 5.2.The scalars ω ℓ and ω m satisfies In particular, the weight function ω is unique up to a scalar multiple.
Proof.By comparing the diagonal entries of both sides in (61), we have Likewise, By eliminating κ e , we obtain the formula for ω m (ω ℓ ) −1 .The uniqueness of ω follows from this and (51).Example 5.6.Suppose that e = 2. Then we .
From Example 3.13 we find two parameter sets satisfying Assumption 5.2: n ℓ m ξ 22 6 7 3, 5 22 6 15 1, 3 The zeros ξ given in the last column are indeed integers.Note that the other two parameter sets in Example 3.13 correspond to the complements of these two; cf.Lemma 5.1.On the other hand, the existence of tight relative 4-designs with the following feasible parameter sets was left open in [9, Section 6]: n ℓ m ξ 37 9 16 Here, we are again taking Lemma 5.1 into account.Observe that the zeros ξ are irrational, thus proving the non-existence.
We end this section with a comment on the expressions of the polynomials ψ ℓ,ℓ e (ξ) and ψ m,m e (ξ).We first invoke the following identity which agrees with the formula of the backward shift operator on the dual Hahn polynomials (cf.[31, Section 1.6]): This can be routinely verified by writing the LHS as a linear combination of the polynomials (1 − ξ) i (0 i r) using and then comparing the coefficients of both sides.Setting α = ℓ − n, β = −ℓ − 1, and N = ℓ − 1 in (68), it follows that the first term of the RHS in (62) is rewritten as follows: Likewise, the first term of the RHS in (64) is given by

Zeros of the Hahn and Hermite polynomials
Recall the Hahn polynomials Q r (ξ; α, β, N ) from (27).Recall also that the zeros of orthogonal polynomials are always real and simple; see, e.g., [42,Theorem 3.3.1].It is well known that we can obtain the Hermite polynomials as limits of the Hahn polynomials; cf.[30,31].In this section, we revisit this limit process and describe the limit behavior of the zeros of the Q r (ξ; α, β, N ), in a special case which is suited to our purpose.Assumption 6.1.Throughout this section, we assume that α < −N and β < −N , so that the Q r (ξ; α, β, N ) satisfy the orthogonality relation (28).We consider the following limit: We write , and assume further that Remark 6.2.We do not require in Assumption 6.1 that α ǫ , β ǫ , and N ǫ are uniquely determined by ǫ.In other words, these are multi-valued functions of ǫ in general (for admissible values of ǫ), but their limit behaviors are uniformly governed by ǫ.
The following is part of the estimates on the zeros of h e (η) used in [1].[42].Since we have .
Hence, for odd e 5 we have (η e 1 ) 2 < 7 2e + 1 (η In this section, we prove that Theorem 7.1.For any δ ∈ (0, 1/2), there exists e 0 = e 0 (δ) > 0 with the property that, for every given integer e e 0 and each constant c > 0, there are only finitely many tight relative 2e-designs (Y, ω) (up to scalar multiples of ω) supported on two shells Our proof is an application of Bannai's method from [1].We will use the following result, which is a variation of [40, Satz I]: Proposition 7.2.For any ϑ > 0 and δ ∈ (0, 1/ϑ), there exists k 0 = k 0 (ϑ, δ) > 0 such that the following holds for every given integer k k 0 and each constant c > 0: for all but finitely many pairs (a, b) of positive integers with b < c • a δ , the product of k consecutive odd integers has a prime factor which is greater than 2k + 1 and whose exponent in this product is greater than that in The proof of Proposition 7.2 will be deferred to the appendix.We will establish Theorem 7.1 by contradiction: Assumption 7.3.We fix δ ∈ (0, 1/2).Let k 0 = k 0 (2, δ) > 0 be as in Proposition 7.2 (applied to ϑ = 2), and set e 0 = e 0 (δ) = max{2k 0 , 8}.
Proof.Since m, n − m ℓ by Assumption 5.2, it suffices to show that ℓ → ∞.Suppose the contrary, i.e., that there is a sequence Since the ℓ k are bounded, it follows from (88) and (89) that there are only finitely many choices for ψ ℓ,m e (ξ)/s 0 when (ℓ, m, n) ranges over this sequence.In particular, there are only finitely many choices for each of the coefficients s 1 /s 0 and s 2 /s 0 , and hence the same is true (cf.( 90 However, it is immediate to see that these distinct scalars in turn determine n and m uniquely, from which it follows that the n k are bounded, a contradiction.Then the r k and the t k are bounded since n k by Claim 1. From (90) it follows that s 1 /s 0 and s 2 /s 0 are bounded as well, and hence take only finitely many non-zero integral values when (ℓ, m, n) ranges over this sequence.It follows that the r k and the t k can assume only finitely many values, and then since r k ≈ t k we must have r k = t k for sufficiently large k.However, it is again immediate to see that r k = t k for every k ∈ N, and hence this is absurd.It follows that the result holds when ρ = 1.
Finally, suppose that ρ = 0.For every (ℓ, m, n) ∈ Θ we have From (90) and Assumption 5.2 it follows that these scalars are non-zero integers.By the same argument as above, but working with these two scalars instead of s 1 /s 0 and s 2 /s 0 , we conclude that the result holds in this case as well.
By Claims 1 and 2, it follows that the parameters α, β, and N from (86) satisfy Assumption 6.1 when f (ℓ, m, n) → (0, ρ), since Note that the scalar ρ in Assumption 6.1 agrees with the one used here in this case.Hence we are now in the position to apply the results of the previous section to ψ ℓ,m e (ξ), which is the Hahn polynomial having these parameters.Claim 3. We have ρ = 1/2.In particular, (0, 1/2) is a unique accumulation point of f (Θ).Moreover, we have n = 2m for all but finitely many (ℓ, m, n) ∈ Θ.
Proof.Let the ξ i be as in (87).Then from Propositions 6.4 and 6.5 it follows that (92) for all i, j, as f (ℓ, m, n) → (0, ρ), where the η i are the zeros of the monic Hermite polynomial h e (η) from (77) as in Proposition 6.4.Recall that e 8 by Assumption 7.3.Set (i, j) = (1, 0) in ( 92) if e is odd, and (i, j) = (2, 1) if e is even.Then, since 4ρ − 2 3 2 3 , it follows from Proposition 6.5 that the RHS in (92) lies in the open interval (−1, 1).However, the LHS in (92) is always an integer by (88), so that this is possible only when the RHS equals zero, i.e., ρ = 1/2.In particular, we have shown that (0, 1/2) is a unique accumulation point of f (Θ).Again by ( 88) and (92), we then have ξ i + ξ −i = ξ j + ξ −j for all i, j, provided that f (ℓ, m, n) is sufficiently close to (0, 1/2).By the uniqueness of the accumulation point and (91), this last condition on f (ℓ, m, n) can be rephrased as "for all but finitely many (ℓ, m, n) ∈ Θ." Now, let ξ be the average of the zeros ξ i of ψ ℓ,m e (ξ).Then the above identity means that the ξ i are symmetric with respect to ξ.Hence, if we write where ν p (n) denotes the exponent of p in n.Assuming that this is the case, let i (1 i k) be such that ν p (2m − 2e + 2i + 1) > 0.
Observe that i is unique since p > 2k + 1, so that we have However, this contradicts the fact that s 2i /s 0 is a non-zero integer.Hence we now conclude that Θ must be finite.
The proof of Theorem 7.1 is complete.
Let W ∈ Λ r , where 0 r e. Recall Theorem 4.3 and also the standard basis(21) of W .If r = e then E e W is spanned by v W , and hence we have ω ℓ = ω, Xℓ , ω m = ω, Xm .
and define S ℓ,m (Y )(= S m,ℓ (Y )) and S m,m (Y ) in the same manner.Let H be the set consisting of the |Y | × |Y | matrices of the form (59)

Let Θ denote the
set of triples (ℓ, m, n) ∈ N 3 taken by those (Y, ω) in Assumption 7.3.Recall from Proposition 5.4 that ω is uniquely determined by Y up to a scalar multiple.Moreover, for each (ℓ, m, n) ∈ Θ there are only finitely many choices for Y .Hence we have (85) |Θ| = ∞.
[53]complement of this is yet another example 1 such that L = {k − 1, n − k}.See[32, Section 3]and[50, Theorem 8].Note that the weights are constant for these three examples.O the other hand, Bannai, Bannai, and Bannai[5, Theorem 2.2]  showed that there is a tight relative 2-design in Q n with L = {2, n/2} for n ≡ 6 (mod 8), provided that a Hadamard matrix of order n/2 + 1 exists.This construction provides examples in which the weights take two distinct values depending on the shells.See also[53].Example 3.13.Working with the tight 4-(23, 7, 1) and 4-(23, 16, 52) designs instead of a symmetric 2-(n + 1, k, λ) design as in Example 3.12, we obtain four tight relative 4-designs in Q 22 with constant weight such that and [6, Theorem 2.7, Example 2.9].Example 3.12.Consider a symmetric 2-(n+ 1, k, λ) design (cf.[13,Chapter II.6]).Observe that removing a point yields a tight relative 2-design in Q n with L = {k − 1, k}.Likewise, taking the complement of every block which contains a given point followed by removing that point gives rise to a tight relative 2-design in Q n with L = {k, n + 1 − k}.