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-Cramp\'{e}-d'Andecy-Vinet-Zaimi [arXiv:2212.10824], and then followed by Bannai-Kurihara-Zhao-Zhu [arXiv:2305.00707]. 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 attenuated space association schemes 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.


Introduction
The study of P -polynomial association schemes (distance-regular graphs) and Q-polynomial association schemes has been one of the central problems in the study of algebraic combinatorics, in particular in the theory of association schemes.The classification of Pand Q-polynomial association schemes was very much looked for, and their connection with the theory of orthogonal polynomials has been very important since it was shown by Leonard [15] that the spherical functions (and character tables) of P -and Q-polynomial association schemes were described by using Askey-Wilson orthogonal polynomials including their special or limiting cases (See [1,4,5,23,24]).Note that the orthogonal polynomials attached to P -and Q-polynomial association schemes are one-variable polynomials.It seems that many researchers wanted to define higher rank P -and Q-polynomial association schemes, but only some special cases have been studied, see Mizukawa-Tanaka [16], Gasper-Rahman [12], Iliev-Terwilliger [13] and many others.On the other hand, onevariable Askey-Wilson orthogonal polynomials have been generalized to multivariable orthogonal polynomials in various ways and also very much studied, purely as orthogonal polynomials.See, for example, many papers mentioned in the Introduction and the References in Bernard-Crampé-d'Andecy-Vinet-Zaimi [4].However, their connection with the theory of association schemes, in particular, higher rank P -and Q-polynomial association schemes, was not very much revealed, except for the special cases studied in the papers referenced above [16,12,13], among others.The reason for this was that the good definition of higher rank (i.e., multivariate) P -polynomial or Q-polynomial association schemes were missing and explicit good examples of higher rank P -polynomial or Q-polynomial association schemes were generally missing.So, it was a very pleasant surprise when we saw the paper by Bernard-Crampé-d'Andecy-Vinet-Zaimi [4], where the bivariate P -polynomial association schemes (and the bivariate Q-polynomial association schemes) of type (α, β) were defined and showed that some good explicit examples do exist.Motivated by their paper [4], we started to think of this line of study, and we defined multivariable P -polynomial (and Q-polynomial) association schemes generalizing their concept for any monomial order.Then, the authors [3] obtained more examples of (family of) multivariate P -polynomial association schemes and also the multivariate Q-polynomial association schemes.
The purpose of the present paper is to show that the two important families are actually Q-polynomial association schemes in the sense of [3] with respect to the graded lexicographic order.In most cases, we consider multivariate P -polynomial (or Q-polynomial) association schemes with respect to the graded lexicographic order.Namely, the main results of the present paper are the following Theorem 1.1 and Theorem 1.2 to be described below.It was shown by Crampé-Vinet-Zaimi-Zhang [8] that nonbinary Johnson association schemes are bivariate Q-polynomial in their sense [4] if some restriction is imposed on the parameters.Our proof is basically along their lines and to calculate the Krein parameters very explicitly.However, we can completely lift the parameter restrictions in [8].Another case was on association schemes coming from attenuated spaces.The parameters and spherical functions of this family of association schemes are obtained by Wang-Guo-Li [25] and Kurihara [14] already, but to show that these association schemes are actually bivariate Q-polynomial is highly non-trivial, as we need to compute the Krein parameters explicitly.We show that they are indeed bivariate Q-polynomial, again lifting the parameter restrictions completely.
Theorem 1.1.The nonbinary Johnson scheme J r (n, k) is a bivariate Q-polynomial association scheme with respect to the graded lexicographic order ≤ grlex .
Theorem 1.2.The association scheme obtained from attenuated spaces is a bivariate Q-polynomial association scheme with respect to the graded lexicographic order ≤ grlex .
We hope and expect that the techniques used in the present paper will be applied to other families, in particular for association schemes on (not necessarily maximal) isotropic spaces (see Stanton [17]) and generalized Johnson association schemes in the sense of Ceccherini-Silberstein, Scarabotti and Filippo [6], but the technical difficulties are so far beyond our ability.We want to come back to this question in the near future.Finally, we note that many examples of our multivariate P -and Q-polynomial association schemes actually become some explicit examples of Problem 7.1 in Iliev-Terwilliger [13].Therefore, we believe this will pave the way for research in new directions.Now we will provide further details regarding the content of this paper.In Section 2, we will review the definitions of multivariate P -or Q-polynomial association schemes in the sense of [3], building upon the work of Bernard et al. [4].Section 3 will present relevant facts concerning nonbinary Johnson schemes.The proof of Theorem 1.1 will be provided in Section 4. Section 5 will discuss association schemes obtained from attenuated spaces.The proof of Theorem 1.2 will be presented in Section 6.Furthermore, in Section 7, we will explore the relationship between nonbinary Johnson schemes, association schemes derived from attenuated spaces, and A 2 -Leonard pairs.
2 Multivariate P -polynomial and/or Q-polynomial association schemes In this section, we briefly review the definition and properties of multivariate P -polynomial and/or Q-polynomial association schemes introduced by Bannai-Kurihara-Zhao-Zhu [3].Please refer to [3] for more details.Note that the definition of bivariate P -polynomial and/or Q-polynomial association schemes was originally introduced by Bernard-Crampéd'Andecy-Vinet-Zaimi [4], and Bannai et al. [3] extended it to the multivariate case.

Association schemes
We begin by recalling the concept of association schemes.The reader is referred to Bannai-Bannai-Ito-Tanaka [1] and Bannai-Ito [2] for details.Let X and I be finite sets and let R be a surjective map from X × X to I.For i ∈ I, we put Let M X (C) be the C-algebra of complex matrices with rows and columns indexed by X.The adjacency matrix A i of i ∈ I is defined to be the matrix in M X (C) whose (x, y) entries are

It is obvious that (A1)
i∈I A i = J X , where J X is the all-one matrix of M X (C).A pair X = (X, {R i } i∈I ) (or simply (X, R)) is called a commutative association scheme if X satisfies the following conditions: (A2) there exists i 0 ∈ I such that A i 0 = I X , where I X is the identity matrix of M X (C); (A3) for each i ∈ I, there exists i ′ ∈ I such that Moreover, if an association scheme X = (X, We also use the notation X = (X, {A i } i∈I ) to denote association schemes with the adjacency matrices {A i } i∈I .Let A = Span C {A i } i∈I .By (A2) and (A4), A becomes a subalgebra of M X (C).The algebra A is called the Bose-Mesner algebra of X.By (A1), {A i } i∈I is a basis of A, i.e., dim C A = |I|.
By (A5), A has another basis {E j } j∈J consisting of the primitive idempotents of A, where J is a finite set and there exists j 0 ∈ J such that E j 0 = 1 |X| J X .Since {A i } i∈I and {E j } j∈J are bases of A, |I| = |J | holds.By (A1), A is closed under entrywise multiplication, which product is denoted by • and called the Hadamard product.Then for i, j, k ∈ J , there exists a real number (in fact, a nonnegative number) and q k ij are called the Krein numbers of X.The entries of the first eigenmatrix P := (P i (j)) j∈J ,i∈I and the second eigenmatrix Q := (Q j (i)) i∈I,j∈J of X are defined by respectively.Note that P i (j) and Q j (i) are complex numbers in general, and if X is symmetric, then P i (j) and Q j (i) are real numbers.It is known that k i := P i (j 0 ) and m j := Q j (i 0 ) are positive integers, and k i and m j are called the valencies and the multiplicities of X, respectively.Also one can check easily that P i 0 (j) = 1 and Q j 0 (i) = 1 hold.Then the following formula of the Krein number holds: for i, j, k ∈ J .Here P l (k) denotes the complex conjugate of P l (k).By P l (i)/k l = Q i (l)/m i , the above formula (2.1) can be rewritten as By q j i,j 0 = δ ij , where δ ij is the Kronecker delta, we have Note that (2.3) is known as the orthogonality relation of the second eigenmatrix Q.
A symmetric association scheme X = (X, {R i } i∈I ) of class d is called P -polynomial if it satisfies the following conditions: I = {0, 1, . . ., d} and there exists a univariate polynomial v i of degree i such that A i = v i (A 1 ) for each i ∈ {0, 1, . . ., d}.Similarly, a symmetric association scheme X = (X, {R i } i∈I ) of class d is called Q-polynomial if it satisfies the following conditions: J = {0, 1, . . ., d} and there exists a univariate polynomial v * j of degree j such that |X|E j = v * j (|X|E 1 ) (under the Hadamard product) for each j ∈ {0, 1, . . ., d}.The following condition is well known as an equivalent condition of the property of P -polynomial: for i ∈ {0, 1, . . ., d}, the three-term recurrence formula holds.Note that p −1 10 A −1 and p d+1 1d A d+1 are regarded as zero.Similarly, an equivalent condition of the property of Q-polynomial is the following: for i ∈ {0, 1, . . ., d}, the three-term recurrence formula holds.Note that q −1 10 |X|E −1 and q d+1 1d |X|E d+1 are regarded as zero.
Definition 2.1.A monomial order ≤ on N ℓ is a relation on the set of N ℓ satisfying: (i) ≤ is a total order; (iii) ≤ is a well-ordering, i.e., any nonempty subset of N ℓ has a minimum element under ≤.
The multivariate Q-polynomial association scheme is defined by replacing the adjacency matrices with the primitive idempotents in the definition of the multivariate P -polynomial association scheme as Definition 2.4.An equivalent condition for the multivariate Q-polynomial association scheme is also given by Proposition 2.5 in the same way as Proposition 2.3.

Definition 2.4 ([3]
).Let D * ⊂ N ℓ having ǫ 1 , ǫ 2 , . . ., ǫ ℓ and ≤ be a monomial order on N ℓ .A commutative association scheme X = (X, R) with the primitive idempotents {E j } j∈J is called ℓ-variate Q-polynomial on the domain D * with respect to ≤ if the following three conditions are satisfied: (ii) there exists a relabeling of the adjacency matrices: . ., ǫ ℓ and X be a commutative association scheme with the primitive idempotents {E α } α∈D * indexed by D * .The statements (i) and (ii) are equivalent: (i) X is an ℓ-variate Q-polynomial association scheme on D * with respect to ≤; (ii) the condition (i) of Definition 2.4 holds for D * and the Krein numbers satisfy, for each i = 1, 2, . . ., ℓ and each α ∈ D * , q

Nonbinary Johnson schemes
In this section, we describe the nonbinary Johnson scheme, which is a generalization of the Hamming schemes and the Johnson schemes, and in this paper, we require various properties of these schemes.Therefore, in Subsection 3.1, we first describe the properties of the Hamming schemes and the Johnson schemes.In Subsection 3.2, we describe the nonbinary Johnson schemes.

Hamming schemes and Johnson schemes
Let H(n, q) be the Hamming scheme, i.e., the set of all q-ary n-tuples with the adjacency relation R i defined by the Hamming distance i.Then the class of H(n, q) is n.For details of the Hamming scheme, see [1,2,5].Let k and m H(n,q) i be the valency and the multiplicity of H(n, q), respectively.These values are known as Also, let p k ij (H(n, q)) and q k ij (H(n, q)) be the intersection number and the Krein number of H(n, q), respectively.It is well known that Hamming schemes are P -and Q-polynomial association schemes.We give a formula for these numbers for later use: Moreover, let P H(n,q) i (j) and Q H(n,q) i (j) be the (i, j)-entry of the first and the second eigenmatrix of H(n, q), respectively.Then we have where K i (n, q; j) is the Krawtchouk polynomial defined by Let J(n, k) be the Johnson scheme, i.e., the set of all k-subsets of {1, 2, . . ., n} with the adjacency relation R i defined by the symmetric difference of cardinality i.Then the class of be the valency and the multiplicity of J(n, k), respectively.These values are known as It is well known that Johnson schemes are P -and Q-polynomial association schemes.Let q γ st (J(n, k)) be the Krein number of the Johnson scheme J(n, k).We give a formula for these numbers for later use: where .
Moreover, let P (j) be the (i, j)-entry of the first and the second eigenmatrix of J(n, k), respectively.Then we have where E i (n, k; j) and H i (n, k; j) are the Eberlein polynomial (dual Hahn) and the Hahn polynomial defined by For the Hahn polynomials, we have the following recurrence relation.This relation is due to [8].
Although (3.6) seems not to hold for some values of r, it can be justified by interpreting it as follows.For r = 0, we regard H −1 (N − 1, p − 1; x) as zero, i.e., (3.6) is written by In fact, both H 0 (N, p; x) and H 0 (N − 1, p − 1; x) become 1, hence the above equation is correct.For r = p, we regard H p (N − 1, p − 1; x) as zero, i.e., (3.6) is written by In fact, by , the above equation is justified.

Nonbinary Johnson schemes
Let n and r be positive integers such that r > 1 and let k be a natural number such that 0 ≤ k ≤ n.Let K = {0, 1, . . ., r − 1} be a set of cardinality r.For a vector x = (x 1 , x 2 , . . ., x n ) ∈ K n , its weight w(x) is defined by the number of non-zero entries, namely w(x) = |{i | x i = 0}|.We consider the following set S = {x ∈ K n | w(x) = k}.Note that the cardinality of S is given by For two vectors x = (x 1 , x 2 , . . ., x n ) and y = (y 1 , y 2 , . . ., y n ) in K n , we consider the number of equal non-zero entries e(x, y) and the number of common non-zero entries c(x, y), namely Then the pair (S, R) becomes a symmetric association scheme, which we call the nonbinary Johnson scheme.We denote it by J r (n, k).Note that if r = 2, then J r (n, k) is the Johnson scheme J(n, k) and if n = k, then J r (n, k) is the Hamming scheme H(n, r).Therefore, we focus on the case where r > 2 and n > k.
It is known that the indices of the primitive idempotents of J r (n, k) are also labeled by D. The explicit expression of eigenvalues of J r (n, k) are given by Tarnanen-Aaltonen-Goethals [19].For (i, j), (x, y) ∈ D, we have Moreover, the valencies of J r (n, k) are given by Similarly, for (i, j), (x, y) ∈ D, the entries of the second eigenmatrix of J r (n, k) are given by Moreover, the multiplicities of J r (n, k) are given by 4 Proof of Theorem 1.1 In order to prove Theorem 1.1, it is sufficient to prove the following proposition from Proposition 2.5.
Remark 1. Proposition 4.1 was essentially already shown by [8].However, the proof is given again using an approach different from [8] in the following.This calculation will help in understanding Section 6.

Krein numbers q st
10,ij To prove (i) of Proposition 4.1, we compute the Krein numbers q st 10,ij of the nonbinary Johnson scheme using the formula (2.2).Note that J r (n, k) is symmetric, so there is no need to consider complex conjugates.By (3.7), (3.9), (3.10) and (3.11), we have x Applying (2.2) to the Hamming scheme H(k − y, r − 1), we have x According to (3.2), s must be i − 1, i or i + 1.In the below argument, we will use When the case s = i in (4.2), by (3.3) and (4.3), we have , the range of y in the above summation will be 0 ), the range of y in the above summation will be 0 ≤ y ≤ (k − i − 1) ∧ (n − k).From Lemma 3.1, In the last line, we used Thus, t must be j or j − 1, and we have Moreover, if (i + 1, j) ∈ D, then (i + 1) + j ≤ k, i.e., k − i − j ≥ 1 and j ≤ n − k, i.e., n − i − 2j ≥ 1.This implies q i+1,j 10,ij = 0.When the case s = i − 1 in (4.2), by ) and (3.3), we have , the range of y in the above summation will be 0 ≤ y ≤ (k − i) ∧ (n − k).From Lemma 3.1, holds.Then the orthogonal relation (2.3) for J(n − i, k − i) implies Thus, t must be j or j + 1.If t = j, then the Krein number becomes By (3.4), the above multiplicities can be calculated as Thus, we have If t = j + 1, the Krein number becomes .
By (3.4), the multiplicity m can be calculated as n − i j .
Thus, we have Applying (2.3) to the Hamming scheme H(k − y, r − 1), we have Hence, s must be i and by (3.1), we have , the range of y in the above summation will be 0 ≤ y ≤ (k − i) ∧ (n − k).To carry out the above summation calculations, we replace H 1 (n, k; y) with another expression.By the definition of the Hahn polynomials, we have ) Eliminating y from (4.9) and (4.10) yields the following equation: In the last line, we used This implies t must be j, j + 1 or j − 1, and we have ; (4.12) This implies q ij+1 01,ij = 0. Note that q ij 01,ij coincides with B ij in [8].Remark 2. J r (n, k) can be regarded as the Gelfand pair A part of the proof of Proposition 4.1 can also be shown using the representation theory of S r−1 ≀ S n .Specifically, the irreducible representations of S r−1 ≀ S n are known from Stein [18], and the irreducible representations of S r−1 ≀ S n that appears in the permutation representation of the above Gelfand pair are determined by the result of Ceccherini-Silberstein, Scarabotti, Tolli [6].Then, using the Littlewood-Richardson rule for S r−1 ≀S n , we can investigate for which (s, t), Krein numbers q st 10,ij and q st 01,ij become zero.However, it is so far difficult to show q i+1,j 10,ij = 0 and q i,j+1 01,ij = 0 using this method.

Association schemes obtained from attenuated spaces
In this section, we describe the association schemes obtained from attenuated spaces, which is a generalization of the bilinear forms schemes and the Grassmann schemes, and in this paper, we require various properties of these schemes.Therefore, in Subsection 5.1, we first describe the properties of the bilinear forms schemes and the Grassmann schemes.In Subsection 5.2, we describe the association schemes obtained from attenuated spaces.In this section, let q be a prime power, and n, m and l be positive integers.

Bilinear forms schemes and Grassmann schemes
Let F q be the finite field of size q, and let V and E be n-dimensional and l-dimensional vector spaces over F q , respectively, and let L(V, E) denote the set of all linear maps from V to E. Then the size of L(V, E) is q nl .The set L(V, E) together with the nonempty relations is an n∧l-class symmetric association scheme called the bilinear forms scheme H q (n, l).It is well known that bilinear forms schemes are P -and Q-polynomial association schemes.The first eigenmatrix P = (P Hq(n,l) i (j)) of H q (n, l) is given by the generalized Krawtchouk polynomials K i (n, l; q; j) (see Delsarte [11]), namely P Hq(n,l) i (j) = K i (n, l; q; j) is called the q-binomial coefficient for m < n.If either i or j is outside {0, 1, . . ., n ∧ l}, then we define K i (n, l; q; j) = 0. Note that bilinear forms schemes are self-dual, i.e., the second eigenmatrix of H q (n, l) coincides with the first eigenmatrix of H q (n, l) (see [11]).Moreover, the valencies k and the multiplicities m (q i − q u ).
(5.1) Also, the Krein numbers q s 1i (H q (m−y, l)) coincide with the intersection numbers p s 1i (H q (m− y, l)) and these numbers satisfy where [k] := (q k − 1)/(q − 1) is the q-number.One can easily check that Assume m < n.The set of all m-dimensional subspaces of V is denoted by V m .The set V m together with the nonempty relations -class symmetric association scheme called the Grassmann scheme Gr q (n, m).It is well known that Grassmann schemes are P -and Q-polynomial association schemes.The size of V m is n m .The first eigenmatrix P = (P Grq(n,m) i (j)) of the Grassmann scheme Gr q (n, m) is given by the generalized Eberlein polynomials E i (n, m; q; j) (see Delsarte [10]), namely If either i or j is outside of {0, 1, . . ., m ∧ (n − m)}, then we define E i (n, m; q; j) = 0. Furthermore, since the valencies k Grq(n,m) i and the multiplicities m Grq(n,m) i of the Grassmann scheme Gr q (n, m) are given as respectively (cf.[1,2,5]).The second eigenmatrix Q = (Q Grq(n,m) i (j)) of the Grassmann scheme Gr q (n, m) is given by the q-Hahn polynomials Q i (n, m; q; j) as follows; If either i or j is outside {0, 1, . . ., m ∧ (n − m)}, then we define Q i (n, m; q; j) = 0.It is known that where [1]).Hence, we have (5.5) Also, we know where

Association schemes based on attenuated spaces
Let us recall the definition of the association schemes obtained from attenuated spaces.Fix an l-dimensional subspace W of the (n + l)-dimensional vector space F n+l q over F q .The corresponding attenuated space associated with F n+l q and W is the collection of all subspaces of F n+l q intersecting trivially with W .For a positive integer m with m ≤ n, let X be the set of m-dimensional subspaces of the attenuated space associated with F n+l q and W .Let where V /W stands for (V + W )/W simply.Then X = (X, R) is a symmetric association scheme, and called the association scheme obtained from the attenuated space associated with F n+l q and W .For details of the association schemes obtained from attenuated spaces, see Bernard et al. [4], Wang-Guo-Li [25] or Kurihara [14].The association scheme X is a common generalization of the Grassmann scheme Gr q (n, m) and the bilinear forms scheme H q (n, l).In fact, if l = 0, then the association scheme X is isomorphic to the Grassmann scheme Gr q (n, m) and if m = n, then the association scheme X is isomorphic to the bilinear forms scheme H q (n, l).Moreover, the association scheme X is also a q-analogue of the nonbinary Johnson scheme.Bernard et al. [4] proved that in the case of l ≥ m, X becomes bivariate P -polynomial of type (1, 0) on the domain D. Subsequently, Bannai et al. [3] showed that X is a bivariate P -polynomial association scheme with respect to ≤ grlex in the sense of Definition 2.2 even if l < m.
The cardinality of X = (X, R) is given by It is known that the indices of the primitive idempotents of X are also labeled by D. By [14], the explicit expressions of the first and second eigenmatrices of X are given by and Moreover, the valencies k ij and the multiplicities m ij of X are given by (5.10) In concluding this subsection, we give some formulas and a lemma that will be used in the next sections.First, the following formula may be easily verified: The following lemma is a q-analog of Lemma 3.1.The proof of the lemma is also similar to [8].
In fact, both Q 0 (N, p; q; x) and Q 0 (N − 1, p − 1; q; x) become 1, hence the above equation is correct.For r = p, we regard Q p (N − 1, p − 1; q; x) as zero, i.e., (5.17) is written by In fact, by , the above equation is justified.

Let us focus on the factor
in each summand and rewrite it using (5.16) and (5.12) as follows: Then by (5.15) and (5.16), we have and by (5.12), ( Therefore, we have According to (5.2), s must be i − 1, i or i + 1 In the below argument, we will use When the case s = i + 1 in (6.1), by (5.3), (5.2) and (6.2) we have ), the range of y in the above summation will be 0 ≤ y ≤ (m − i − 1) ∧ (n − m).From (5.17), In the last line, we used Thus, t must be j or j − 1, and we have When the case s = i − 1 in (6.1), by ) and (5.3), we have , the range of y in the above summation will be 0 ≤ y ≤ (m − i) ∧ (n − m).From Lemma 5.1, holds.The orthogonal relation (2.3) for Gr q (n − i, m − i) implies Thus, t must be j or j + 1. Assume t = j.Then the Krein number becomes By (5.4) and (5.16), the above multiplicities can be calculated as n − i j .
Thus, we have Assume t = j + 1.Then the Krein number becomes .
By (5.4) and (5.16), the multiplicity m can be calculated as n − i j .
Applying (2.3) to H q (m − y, l), we have x K i (m − y, l; q; x)K s (m − y, l; q; x) = q (m−y)l m Hq(m−y,l) s δ i,s .
Hence, s must be i and by (5.3), (5.1) and (6.2), we have , the range of y in the above summation will be 0 ≤ y ≤ (m − i) ∧ (n − m).To carry out the summation, we replace Q 1 (n, m; q; y) with another expression.Similar way to (4.11), we have In order to compute the sum, we use the following calculation: holds.By (5.6), t must be j − 1, j or j + 1.When t = j − 1, we have .11)When t = j + 1, we have When t = j, we have . (6.13)

A M -Leonard pairs
Iliev-Terwilliger [13] consider some multivariate P -polynomial (and/or Q-polynomial) association schemes from the viewpoint of root systems, in particular of type A n and possibly for other types.These are very special classes of more general multivariate Ppolynomial (and/or Q-polynomial) association schemes we have considered.Recently, Crampé-Zaimi [9] gave further progress with respect to the above theory of [13].Here we give the definition of the generalized Lenard pair, called the A M -Leonard pair, described in [9].Let F denote a field and let V denote a vector space over F with finite positive dimension.Let End(V ) denote the F-algebra consisting of the F-linear maps from V to V .Fix integers M, N ≥ 1.Let A pair of elements (r 1 , . . ., r M ) and (r ′ 1 , . . ., r ′ M ) in D will be called adjacent whenever (r 1 − r ′ 1 , . . ., r M − r ′ M ) is a permutation of an element in {(0, 0, . . ., 0), (1, −1, 0, 0, . . ., 0), (1, 0, 0, . . ., 0), (−1, 0, 0, . . ., 0)}.(i) H is an M-dimensional subspace of End(V ) whose elements are diagonalizable and mutually commute.
(ii) H is an M-dimensional subspace of End(V ) whose elements are diagonalizable and mutually commute.(v) There does not exist a subspace W of V such that HW ⊂ W , HW ⊂ W , W = 0, W = V .
Among multivariate polynomial association schemes, we define a class related to the A M Leonard pairs and show that it has the structure of A M -Leonard pairs as follows.
Definition 7.2.A symmetric association scheme X is called an A M multivariate P -and Q-polynomial association scheme on D = {α ∈ N M | |α| ≤ N} if the following conditions are satisfied: (i) X is an M-variate P -polynomial association scheme on D for some monomial order ≤ 1 such that for α ∈ D and i = 1, 2, . . ., M, if p β ǫ i ,α = 0 then β is adjacent to α; (ii) X is an M-variate Q-polynomial association scheme on D for some monomial order ≤ 2 such that for α ∈ D and i = 1, 2, . . ., M, if q β ǫ i ,α = 0 then β is adjacent to α.
By considering the principal module of the Terwilliger algebra of an association scheme, it can be shown that an A M -multivariate P -and Q-polynomial association scheme has the structure of A M -Leonard pairs as Theorem 7.3.We will now briefly review the principal module of the Terwilliger algebra.For details, please refer to Terwilliger [20,21,22].
Let X = (X, {R i } i∈I ) be a general commutative association scheme.The complex vector space with the set X of points is represented by W = C X .Suppose x 0 ∈ X is a point of X and for each i ∈ I, Γ i (x 0 ) = {x | (x 0 , x) ∈ R i }.The subspace of W spanned by Γ i (x 0 ) is denoted as W * i .Let E * i be the projection matrix from W to W * i .In other words, E * i is the diagonal matrix given by E * i (x, y) = 1, if x = y and (x 0 , x) ∈ R i , 0, otherwise.(7.1) For i ∈ J , we define 2) The Bose-Mesner algebra A is generated by {A 0 , A 1 , . . ., A d } and the dual Bose-Mesner algebra A * with respect to x 0 is generated by {A * 0 , A * 1 , . . ., A * d }.The Terwilliger algebra T = T (x 0 ) is generated by A and A * .Let 1 ∈ C X be the all-one vector.It is known that the subspace Ax 0 = A * 1 is T -invariant, and thus Ax 0 is called the principal T -module.
Then one can check the following facts: (T 1) A j v * i = P j (i)v * i for j ∈ I and i ∈ J ; (T 2) A * j v i = Q j (i)v i for i ∈ I and j ∈ J ; (T 3) dim V i = 1 for i ∈ J ; (T 4) dim V * i = 1 for i ∈ I; (T 5) A j v i = k∈I p k j,i v k ; (T 6) A * j v * i = k∈J q k j,i v * k .Theorem 7. Proof.Let H be the subspace spanned by A ǫ 1 , A ǫ 2 , . . ., A ǫ M , and let H be the subspace spanned by A * ǫ 1 , A * ǫ 2 , . . ., A * ǫ M .Then (T 1) and (T 2) lead to (i) and (ii) of Definition 7.1.Let Ṽα = V * α .By (T 3), (T 4), (T 5) and (T 6), and the definition of A M multivariate P -and Q-polynomial, we know that (iii), (iv), and (vi) of Definition 7.1 hold.Since Definition 7.2 requires that an A M multivariate P -and Q-polynomial association scheme is symmetric, we know that (vii) of Definition 7.1 holds as well.
Corollary 7.4.If the domain D of a nonbinary Johnson scheme or an association scheme obtained from attenuated space becomes an isosceles right triangle, then the association scheme has the structure of A 2 -Leonard pairs.

Definition 7 . 1 .
The pair (H, H) is an A M -Leonard pair on the domain D if the following statements are satisfied:

(
iii) There exists a bijection α → V α from D to the set of common eigenspaces of H such that for all α ∈ D,HV α ⊂ β∈D β adj α V β .(iv)There exists a bijection α → Ṽα from D to the set of common eigenspaces of H such that for all α ∈ D, H Ṽα ⊂ β∈D β adj α Ṽβ .

3 .
An A M multivariate P -and Q-polynomial association scheme on D = {α ∈ N M | |α| ≤ N} has the structure of A M -Leonard pairs on F = C.