Weak metric and weak cometric schemes

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J. Korean Math. Soc. 46 (2009), No. 4, pp. 785–812 DOI 10.4134/JKMS.2009.46.4.785

WEAK METRIC AND WEAK COMETRIC SCHEMES

Dae San Kim and Gil Chun Kim

Reprinted from the

Journal of the Korean Mathematical Society Vol. 46, No. 4, July 2009

c

2009 The Korean Mathematical Society

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DOI 10.4134/JKMS.2009.46.4.785

WEAK METRIC AND WEAK COMETRIC SCHEMES

Dae San Kim and Gil Chun Kim

Abstract. The notion of weak metric and weak cometric schemes are introduced as a generalization of metric and cometric schemes. They are given as the wreath product of a finite number of symmetric association schemes satisfying certain equivalent conditions which are analogous to the ones for metric or cometric schemes. We characterize those schemes and determine some of their parameters.

1. Introduction

Let F q be the finite field with q elements, and let P = ([n], ≤) be a poset on the underlying set [n] = {1, 2, . . . , n} of coordinate positions of vectors in F ⁿ q . Then the P-weight w _P is the function on F ⁿ _q which is given by

w _P (x) = |{i ∈ [n] | i ≤ j for some j ∈ Supp(x)}|.

Here Supp(x) = {j ∈ [n] | x j 6= 0} for x = (x 1 , . . . , x n ) ∈ F ⁿ _q . Now, d _P (x, y) = w _P (x − y) is a metric, called P-metric. If P is an antichain, P-weight and P -metric reduce respectively to Hamming weight and Hamming metric. The notion of P-codes, namely subsets C ⊆ F ⁿ q equipped with w _P , were introduced in [4] by Brualdi et. al.

For each linear code C ⊆ F ⁿ _q , the P-weight distribution of C is {A _{P, i} (C)} ⁿ _i=0 , where

A _{P, i} (C) = |{x ∈ C | w _P (x) = i }|.

We will denote by Aut(F ⁿ _q , w _P ) the group of all linear automorphisms τ : F ⁿ _q → F ⁿ _q satisfying w _P (τ x) = w _P (x) for all x ∈ F ⁿ q . Let P 0 = n 1 1 ⊕ · · · ⊕ n t 1 be the poset (P 0 is called a weak order poset) given as the ordinal sum of the antichains n i 1 on the set {n 1 + · · · + n i−1 + 1, . . . , n 1 + · · · + n i−1 + n i } for i = 1, . . . , t, i.e., the underlying set is [n] (n = n 1 + · · · + n t ) and the order relation is given by:

i < j ⇔ i ∈ n l 1, j ∈ n m 1 for some l < m.

Received October 19, 2007.

2000 Mathematics Subject Classification. 05E30, 94B60.

Key words and phrases. weak metric scheme, weak cometric scheme.

This work was supported by grant No.R01-2006-000-11176-0 from the Basic Research Program of the Korea Science and Engineering Foundation.

c

2009 The Korean Mathematical Society

785

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Then we have the following fundamental result.

Theorem A. The following are equivalent.

(1) P is a weak order poset on [n].

(2) (F ⁿ _q , {R i } ⁿ _i=0 ), with (x, y) ∈ R i ⇔ d _P (x, y) = i (0 ≤ i ≤ n) is a symmetric translation association scheme.

(3) The P-weight distribution {A _{P, i} (C)} ⁿ _i=0 of C uniquely determines ˘ P - weight distribution {A P, i ˘ (C ^⊥ )} of C ^⊥ for any linear code C ⊆ F ⁿ q . (4) The group Aut(F ⁿ _q , w _P ) acts transitively on each P-sphere S _P (r) = {x ∈

F ⁿ _q |w _P (x) = r} for 0 ≤ r ≤ n.

(1)⇔(2) is shown in [8]. (1)⇒(3) is proved in [5] and [7], and (3)⇒(1) in [7]. Finally, (1) ⇒ (4) is verified in [5] and (4)⇒(1) in [6]. Here we remark that the proofs for (1)⇒(3) were found in the form of Macwilliams-type identi- ties by applying the discrete Poisson summation formula to suitable P-weight enumerators (cf. [5-7]).

Let us now pay our attention to the equivalence (1)⇔(2) in Theorem A.

We will denote the association scheme (F ⁿ _q , {R i } ⁿ _i=0 ), with (x, y) ∈ R i ⇔ d _P

0

(x, y) = i (P 0 = n 1 1 ⊕ n 2 1 ⊕ · · · ⊕ n t 1 ), by H(n 1 , . . . , n t ; q), which is what we call a weak Hamming scheme. Note that this becomes the usual Hamming scheme when t = 1. Recall the following theorem of Delsarte [3], which is usually called generalized MacWilliams identity.

Theorem B. Let Y be an additive code of the translation association scheme X = (X, {R i } ⁿ _i=0 ). Then

(a j (Y ^◦ )) ⁿ _j=0 = 1

|Y | (a i (Y )) ⁿ _i=0 (q ij ).

Here Y is just a subgroup of X, Y ^◦ is the additive code of the dual scheme X ^∗ = (X ^∗ , {R ^∗ _i } ⁿ _i=0 ) of X, given by

Y ^◦ = {χ ∈ X ^∗ | χ(x) = 1 for all x ∈ Y }, a i (Y ) = |{y ∈ Y | (0, y) ∈ R i }|,

a i (Y ^◦ ) = |{χ ∈ Y ^◦ | (1, χ) ∈ R _i ^∗ }|, and (q ij ) is the Q-matrix of the scheme X.

An immediate consequence of Theorem B is the classical MacWilliams identity which can be expressed in weight-enumerator-free form as: for any linear code C ⊆ F ⁿ _q ,

(a j (C ^⊥ )) ⁿ _j=0 = 1

|C| (a i (C)) ⁿ _i=0 (p j (i)),

where p j (x) = p j (x; n, q) is the Krawtchouk polynomial defined by p j (x) =

X j l=0

(−1) ^l (q − 1) ^j−l x l

n − x j − l

(0 ≤ j ≤ n).

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Likewise, we would have yet another way of obtaining “MacWilliams-type identity” if we could find the Q-matrix for the scheme H(n 1 , . . . , n t ; q), which is an example of weak metric schemes, i.e., a finite wreath product of metric schemes (cf. [1], [9]).

Indeed, we observe that, for i = 1, . . . , t, 1 ≤ i 0 ≤ n i , or i = 1, i 0 = 0, d _P

0

(x, y) = n 1 + · · · + n i−1 + i 0 ⇔ x i+1 = y i+1 , . . . , x t = y t , d H (x i , y i ) = i 0 , where d H is the Hamming metric. So identifying F ⁿ _q with F ⁿ _q

¹

× · · · × F ⁿ _q

^t

by writing the elements x ∈ F ⁿ _q as blocks of coordinates x = (x 1 , . . . , x t ) ∈ F ⁿ _q

¹

× · · · × F ⁿ q

^t

, we see that H(n 1 , . . . , n t ; q) = H(n 1 , q) ≀ · · · ≀ H(n t , q) is noth- ing but the wreath product of the Hamming schemes H(n 1 , q), . . . , H(n t , q), which are metric schemes as is well-known. This motivates our study of weak metric and weak cometric schemes. In particular, this will contribute to better understanding of the important weak Hamming scheme H(n 1 , . . . , n t ; q) (cf.

Theorem A above).

This paper is organized as follows. In Section 2, we will fix some notations that will be used throughout this paper. In Section 3, the notion of weak metric schemes is introduced as a finite wreath product of metric schemes.

Several equivalent conditions for being weak metric schemes are presented in Theorem 2. These include the conditions on the shape of the “first intersection matrices for each level”, some modified polynomial relations on the adjacency matrices for each level and some modified polynomial relations on the p-numbers for each level. In Section 4, for schemes given as a finite wreath product of (not necessarily metric) symmetric association schemes, some of their parameters are determined. But, for Proposition 9(b), Lemma 10 and Theorem 11, we assume that the schemes are weak metric schemes. Note that, for weak metric schemes, Theorem 7(a) completely determines the first intersection matrices for each level, whereas Theorem 2 (b) gives only information about the shapes of those ones. Theorem 11 is an analogue of [2, Theorem 1.3, p. 197 ]. However, proving that requires considerably more work. In Section 5, the notion of weak cometric schemes, which is dual to weak metric schemes, is introduced and some equivalent conditions for being cometric schemes are presented. Finally, in Section 6, we will give an example illustrating modified polynomial relations in the case of a weak Hamming scheme.

2. Preliminaries and notations

Let n 1 , . . . , n t be positive integers with n = n 1 + · · · + n t . For each i = 1, 2, . . . , t, let X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

) be a symmetric association scheme. Here we will always assume that the relations of X ⁽ⁱ⁾ are ordered as R ⁽ⁱ⁾ ₀ , R ⁽ⁱ⁾ ₁ , . . ., R ⁽ⁱ⁾ n

_i

. Then X = X ⁽¹⁾ ≀ · · · ≀ X ^(t) = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) is the wreath product of X ⁽¹⁾ , X ⁽²⁾ , . . . , X ^(t) , so that, for (x 1 , . . . , x t ), (y 1 , . . . , y t ) ∈ X,

(x, y) ∈ R n

_i−1

+i

0

⇔ x i+1 = y i+1 , . . . , x t = y t and (x i , y i ) ∈ R _i ⁽ⁱ⁾

₀

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for i = 1, . . . , t, 1 ≤ i 0 ≤ n i , or i = 1, i 0 = 0 (cf. [1], [9]). Here we will always assume that the relations of X are ordered as R 0 , R 1 , . . . , R n .

For each i = 1, . . . , t, let Γ i = (X i , R ⁽ⁱ⁾ ₁ ) be the graph with the distance function ∂ i . Then, for x = (x 1 , . . . , x t ), y = (y 1 , . . . , y t ) ∈ X, we define

d(x, y) =

0, if x = y,

n i−1 + ∂ i (x i , y i ), if x i 6= y i and x i+1 = y i+1 , . . . , x t = y t , (1)

where n i−1 = n 1 + · · · + n i−1 . Then d is a distance function. For this, we only need to check d(x, y) ≤ d(x, z) + d(z, y) for x 6= y. Assume x i 6= y i , x i+1 = y i+1 , . . . , x t = y t . Then x i 6= z i or z i 6= y i , so that

d(x, z) + d(z, y) ≥ n i−1 + ∂ i (x i , z i ) + ∂ i (z i , y i )

≥ n i−1 + ∂ i (x i , y i )

= d(x, y).

Remark 1. (x, y) ∈ R n

_i−1

+1 ⇔ d(x, y) = n i−1 + 1 (i = 1, . . . , t).

For elementary facts about association schemes, one is referred to [2] and [3]. Throughout this paper, the following notations will be used.

• n 1 , . . . , n t positive integers with n 1 + · · · + n t = n, n 1 + · · · + n i = n i

(1 ≤ i ≤ t), n 0 = 0, n t + · · · + n i = n i (1 ≤ i ≤ t), n t+1 = 0.

• X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

) a symmetric association scheme with a fixed ordering R ⁽ⁱ⁾ ₀ , R ⁽ⁱ⁾ ₁ , . . . , R ⁽ⁱ⁾ n

i

of relations (i = 1, . . . , t).

• Some of the parameters of X ⁽ⁱ⁾ used are: valencies v ₀ ⁽ⁱ⁾ = 1, v ⁽ⁱ⁾ ₁ , . . ., v ⁽ⁱ⁾ n

i

, multiplicities m ⁽ⁱ⁾ ₀ = 1, m ⁽ⁱ⁾ ₁ , . . ., m ⁽ⁱ⁾ n

i

, intersection numbers p ^(i)k _jl , Krein parameters q ^(i)k _jl , the adjacency matrices A ⁽ⁱ⁾ ₀ = I, A ⁽ⁱ⁾ ₁ , . . . , A ⁽ⁱ⁾ n

_i

, the irreducible idempotents E ₀ ⁽ⁱ⁾ = |X i | ⁻¹ J, E ₁ ⁽ⁱ⁾ , . . . , E n ⁽ⁱ⁾

_i

, the first intersection matrix L ⁽ⁱ⁾ ₁ = (p ^(i)k _1l ) 0≤k, l≤n

i

, the first dual intersection matrix M ₁ ⁽ⁱ⁾ = (q ^(i)k _1l ) 0≤k, l≤n

_i

.

• X = (X = X 1 ×· · ·×X t , {R j } ⁿ _j=0 ) the wreath product X = X ⁽¹⁾ ≀· · ·≀X ^(t) of X ⁽¹⁾ , . . . , X ^(t) , with a fixed ordering R 0 , R 1 , . . . , R n of relations, so that (x, y) ∈ R n

_i−1

+i

0

⇔ x i+1 = y i+1 , . . . , x t = y t and (x i , y i ) ∈ R ⁽ⁱ⁾ _i

₀

for i = 1, . . . , t, 1 ≤ i 0 ≤ n i , or i = 1, i 0 = 0.

• Some of the parameters of X used are: p-numbers p jl , q-numbers q jl , intersection numbers p ^k _jl , Krein parameters q _jl ^k , valencies v 0 = 1, v 1 , . . . , v n , multiplicities m 0 = 1, m 1 , . . . , m n , the adjacency matrices A 0 , A 1 , . . . , A n , the irreducible idempotents E 0 , E 1 , . . . , E n , intersection matrices L j = (p ^k _jl ) 0≤k, l≤n , dual intersection matrices M j = (q ^k _jl ) 0≤k, l≤n .

• Further notations for the parameters of X are:

θ ⁽ⁱ⁾ _j = p j, n

_i−1

+1 , v ⁽ⁱ⁾ = v n

_i−1

+1 (v ⁽ⁱ⁾ = θ ₀ ⁽ⁱ⁾ ),

ω ⁽ⁱ⁾ _j = q j, n

_i+1

+1 , m ⁽ⁱ⁾ = m n

_i+1

+1 (m ⁽ⁱ⁾ = ω ⁽ⁱ⁾ ₀ ),

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L ⁽ⁱ⁾ = (p ^k _n

_i−1

_{+1, l} ) 0≤k, l≤n the “first intersection matrix of level i”, M ⁽ⁱ⁾ = (q ^k _n

_i+1

_{+1, l} ) 0≤k, l≤n the “first dual intersection matrix of level i”,

[L ⁽ⁱ⁾ ] = (p ^k _n

_i−1

_{+1, l} ) n

_i−1

≤k, l≤n

i

a submatrix of L ⁽ⁱ⁾ , [M ⁽ⁱ⁾ ] = (q ^k _n

_i+1

_{+1, l} ) n

i+1

≤k, l≤n

i

a submatrix of M ⁽ⁱ⁾ .

• Γ i = (X i , R ⁽ⁱ⁾ ₁ ) the graph with distance function ∂ i , d the function on X defined by (1).

3. Weak metric schemes

Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be the symmetric association scheme which is given as the wreath product of the symmetric association schemes X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

) (cf. Section 2). Then it will be called a weak metric scheme if X satisfies further the following equivalent conditions.

Theorem 2. The following are equivalent.

(a) Γ i = (X i , R ⁽ⁱ⁾ ₁ ) is distance-regular for i = 1, . . . , t, and (x, y) ∈ R j ⇔ d(x, y) = j for j = 0, . . . , n.

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(b) For i = 1, . . . , t, we have the following:

(i) The submatrix [L ⁽ⁱ⁾ ] = (p ^k _n

i−1

+1, l ) n

_i−1

≤k, l≤n

_i

of L ⁽ⁱ⁾ = (p ^k n

_i−1

+1, l ) 0≤k, l≤n

is a tridiagonal matrix with nonzero off-diagonal entries.

Moreover, L ⁽ⁱ⁾ has the following entries:

(ii) (p ^k _n

i−1

+1, l ) n

_i

+1≤k, l≤n = v ⁽ⁱ⁾ I, (iii) p ^k _n

i−1

+1, n

_i−1

+1 = v ⁽ⁱ⁾ for 0 ≤ k ≤ n i−1 , (iv) p ⁿ _n

ⁱ⁻¹

⁺¹

i−1

+1, l = v l for 0 ≤ l ≤ n i−1 ,

(v) All the other entries not appearing in (i)-(iv) are zeros.

(c) There are polynomials with real coefficients ψ ₀ ⁽¹⁾ (x) of degree 0, ψ ⁽ⁱ⁾ _k (x) of degree k for i = 1, . . . , t, 1 ≤ k ≤ n i , with ψ ⁽ⁱ⁾ _k (0) = 0 (2 ≤ i ≤ t), and real numbers α ⁽¹⁾ ₀ = · · · = α ⁽¹⁾ n

1

= 0 ,α ⁽²⁾ ₁ , . . . , α ⁽²⁾ n

2

,. . ., α ^(t) ₁ , . . . , α ^(t) n

_t

such that

A n

_i−1

+i

0

= ψ ⁽ⁱ⁾ _i

₀

(A n

_i−1

+1 ) + α ⁽ⁱ⁾ _i

₀

(A 0 + · · · + A n

_i−1

) (3)

for i = 1, . . . , t, 1 ≤ i 0 ≤ n i , or i = 1, i 0 = 0.

(d) p-numbers satisfy

p j, n

_i−1

+i

0

= ψ ⁽ⁱ⁾ _i

₀

(θ ⁽ⁱ⁾ _j ) + α ⁽ⁱ⁾ _i

₀

(p j0 + · · · + p j, n

_i−1

) (0 ≤ j ≤ n),

where ψ _i ⁽ⁱ⁾

₀

’s and α ⁽ⁱ⁾ _i

₀

’s are the same as in (c), and θ _j ⁽ⁱ⁾ = p j, n

_i−1

+1 .

(e) X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

) is a metric scheme for i = 1, . . . , t.

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Proof. We first note the following: for i = 1, . . . , t, 1 ≤ i 0 ≤ n i , or i = 1, i 0 = 0, A n

_i−1

+i

0

A l = v l A n

_i−1

+i

0

for 0 ≤ l ≤ n i−1 .

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For this, we need to see: for 1 ≤ j < i, 1 ≤ k ≤ n j , or j = 1, k = 0, A n

_i−1

+i

0

A n

_j−1

+k = v n

_j−1

+k A n

_i−1

+i

0

.

Let (x, y) ∈ R m . If (x, z) ∈ R n

_i−1

+i

0

, (z, y) ∈ R n

_j−1

+k for some z ∈ X, then (x, y) ∈ R n

_i−1

+i

0

, and hence m = n i−1 + i 0 . On the other hand, if (x, y) ∈ R n

_i−1

+i

0

, then

{z ∈ X| (x, z) ∈ R n

_i−1

+i

0

, (z, y) ∈ R n

_j−1

+k }

= {z ∈ X| (z, y) ∈ R n

_j−1

+k }.

This shows (4).

(a) ⇒ (b) (i) Let k = n i−1 + k 0 , l = n i−1 + l 0 , with 0 ≤ k 0 , l 0 ≤ n i . Suppose p ^k _n

_i−1

_{+1, l} 6= 0. Then d(x, y) = k, d(x, z) = n i−1 +1, d(z, y) = l for some x, y, z.

Then ∂ i (x i , y i ) = k 0 , ∂ i (x i , z i ) = 1, ∂ i (z i , y i ) = l 0 . So |k 0 − l 0 | ≤ 1, and hence

|k − l| ≤ 1. Thus p ^k _n

_i−1

_{+1, l} = 0, if |k − l| ≥ 2, with n i−1 ≤ k, l ≤ n i . Also, p ^k _n

_i−1

_{+1, l} 6= 0, if |k−l| = 1. For any x i , y i ∈ X i , with ∂ i (x i , y i ) = k 0 (≥ 1), there is a path x i = z ⁽ⁱ⁾ ₀ , z ₁ ⁽ⁱ⁾ , . . . , z _k ⁽ⁱ⁾

₀

= y i . Then ∂ i (x i , z ₁ ⁽ⁱ⁾ ) = 1, ∂ i (z ₁ ⁽ⁱ⁾ , y i ) = k 0 − 1.

Choose any points x j ∈ X j for all j 6= i, and set x = (x 1 , . . . , x i−1 , x i , x i+1 , . . . , x t ), y = (x 1 , . . . , x i−1 , y i , x i+1 , . . . , x t ), z = (x 1 , . . . , x i−1 , z ₁ ⁽ⁱ⁾ , x i+1 , . . . , x t ).

Then d(x, y) = n i−1 + k 0 , d(x, z) = n i−1 + 1, d(z, y) = n i−1 + k 0 − 1. So p ^k _n

i−1

+1, k−1 6= 0 for n i−1 +1 ≤ k ≤ n i . Similarly, one shows that p ^k _n

i−1

+1, k+1 6=

0 for n i−1 ≤ k ≤ n i − 1.

(ii) Let n i + 1 ≤ l ≤ n. For any x, y with d(x, y) = l,

p ^l n

_i−1

+1, l = |{z| d(x, z) = n i−1 + 1, d(z, y) = l}|

= |{z| d(x, z) = n i−1 + 1}|

= v ⁽ⁱ⁾ .

Indeed, if l = n j + l 0 (1 ≤ l 0 ≤ n j+1 ) for some j with i ≤ j ≤ t − 1, then, for z with d(x, z) = n i−1 + 1, x j+1 = z j+1 , ∂ j+1 (x j+1 , y j+1 ) = l 0 , z j+2 = y j+2 , . . . , z t = y t , and hence d(z, y) = l is automatic. In the same manner, one shows p ^k _n

i−1

+1, l = 0 for all k, l with n i + 1 ≤ k, l ≤ n, k 6= l.

(iii), (iv) These can be shown similarly to the proof of (ii).

(v) As p ^k _n

i−1

+1, l = 0 ⇔ p ^l _n

i−1

+1, k = 0, we only need to see:

p ^k _n

_i−1

_{+1, l} = 0 for 0 ≤ k ≤ n (k 6= n i−1 + 1), 0 ≤ l ≤ n i−1 ,

(5)

(8)

and

p ^k _n

i−1

+1, l = 0 for n i−1 + 1 ≤ k ≤ n i , n i + 1 ≤ l ≤ n.

Both of these are easy to see.

(b) ⇒ (a) First, we show (2) by induction on j. It is clearly true for j = 0. Assume that k ≥ 1 and the assertion holds for all j ≤ k − 1. Let k = n i−1 + k 0 for some i, k 0 with 1 ≤ i ≤ t, 1 ≤ k 0 ≤ n i . Let (x, y) ∈ R k . As p ^k _n

_i−1

_{+1, k−1} 6= 0, (x, z) ∈ R n

_i−1

+1 , (z, y) ∈ R k−1 for some z. By induction hypothesis d(z, y) = k − 1, and d(x, z) = n i−1 + 1, by Remark 1. Thus

∂ i (z i , y i ) = k 0 − 1, z i+1 = y i+1 , . . . , z t = y t ;

∂ i (x i , z i ) = 1, x i+1 = z i+1 , . . . , x t = z t .

So ∂ i (x i , y i ) ≤ k 0 , x i+1 = y i+1 , . . ., x t = y t , and hence d(x, y) ≤ k. Also, we have d(x, y) ≥ k, since otherwise we would have (x, y) ∈ R j for some j ≤ k − 1, by induction hypothesis. Conversely, assume that d(x, y) = k.

Here, in view of Remark 1, we may let d(x, y) = k = n i−1 + k 0 for some i and k 0 with 2 ≤ k 0 ≤ n i . Then we must have (x, y) ∈ R j for some j ≥ k.

We now exclude the possibility that (x, y) ∈ R j for some j ≥ k + 1. Assume, on the contrary, that (x, y) ∈ R j for some j ≥ k + 1. Then there is a path x i = z ₀ ⁽ⁱ⁾ , z ₁ ⁽ⁱ⁾ , . . . , z ⁽ⁱ⁾ _k

₀

₋₁ , z _k ⁽ⁱ⁾

₀

= y i , and x i+1 = y i+1 , . . . , x t = y t . Put

z = (x 1 , . . . , x i−1 , z ⁽ⁱ⁾ _k

₀

₋₁ , x i+1 , . . . , x t ).

Then d(y, z) = n i−1 +1 ⇒ (y, z) ∈ R n

_i−1

+1 , by Remark 1, and d(z, x) = k−1 ⇒ (z, x) ∈ R k−1 . So p ^j _n

i−1

+1, k−1 6= 0. As j ≥ k + 1 ≥ n i−1 + 3, p ^j _n

i−1

+1, k−1 = 0, by our assumption on the matrix L ⁽ⁱ⁾ . This is a contradiction.

Next, we show that each Γ i is distance-regular. Let x i , y i , x ^′ i , y i ^′ ∈ X i , with

∂ i (x i , y i ) = ∂ i (x ^′ _i , y ^′ _i ) = k 0 (0 ≤ k 0 ≤ n i ). Then we must show:

(6) |{u i ∈ X i | ∂ i (x i , u i ) = l 0 , ∂ i (u i , y i ) = m 0 }|

= | {u i ∈ X i | ∂ i (x ^′ _i , u i ) = l 0 , ∂ i (u i , y ^′ _i ) = m 0 }|

for all l 0 , m 0 with 0 ≤ l 0 , m 0 ≤ n i . This is clearly true for l 0 = 0 or m 0 = 0.

Thus we assume l 0 , m 0 ≥ 1. Choose z j ∈ X j for all j 6= i, and set

˜

x = (z 1 , . . . , z i−1 , x i , z i+1 , . . . , z t ), ˜ y = (z 1 , . . . , z i−1 , y i , z i+1 , . . . , z t ),

˜

x ^′ = (z 1 , . . . , z i−1 , x ^′ _i , z i+1 , . . . , z t ), ˜ y ^′ = (z 1 , . . . , z i−1 , y ^′ _i , z i+1 , . . . , z t ).

Then d(˜ x, ˜ y) = d(˜ x ^′ , ˜ y ^′ ). By (2), we have:

|{u ∈ X| d(˜ x, u) = n i−1 + l 0 , d(u, ˜ y) = n i−1 + m 0 }|

= |{u ∈ X| d(˜ x ^′ , u) = n i−1 + l 0 , d(u, ˜ y ^′ ) = n i−1 + m 0 }|, which implies

|X 1 | · · · |X i−1 | · | {u i ∈ X i | ∂ i (x i , u i ) = l 0 , ∂ i (u i , y i ) = m 0 }|

(9)

= |X 1 | · · · |X i−1 | · | {u i ∈ X i | ∂ i (x ^′ i , u i ) = l 0 , ∂ i (u i , y ^′ i ) = m 0 }|.

So we get (6).

(b) ⇒ (c) Here we denote the entries of the submatrix [L ⁽ⁱ⁾ ] of L ⁽ⁱ⁾ just as in (12) for i = 1, . . . , t. Define the polynomials ψ ⁽¹⁾ _i (x) of degree i (0 ≤ i ≤ n 1 ) recursively by:

ψ ₀ ⁽¹⁾ (x) = 1, ψ ⁽¹⁾ ₁ (x) = x,

xψ _i ⁽¹⁾ (x) = b ⁽¹⁾ _i−1 ψ _i−1 ⁽¹⁾ (x) + a ⁽¹⁾ _i ψ _i ⁽¹⁾ (x) + c ⁽¹⁾ _i+1 ψ ⁽¹⁾ _i+1 (x) (i ≥ 1).

Also, define

α ⁽¹⁾ ₀ = α ⁽¹⁾ ₁ = · · · = α ⁽¹⁾ n

1

= 0.

Then

A i

0

= ψ _i ⁽¹⁾

₀

(A 1 ) + α ⁽¹⁾ _i

₀

A 0 for 0 ≤ i 0 ≤ n 1 . Let i (2 ≤ i ≤ t) be fixed. Define

ψ ₁ ⁽ⁱ⁾ (x) = x, α ⁽ⁱ⁾ ₁ = 0.

Then

A n

_i−1

+1 = ψ ⁽ⁱ⁾ ₁ (A n

_i−1

+1 ) + α ⁽ⁱ⁾ ₁ (A 0 + · · · + A n

_i−1

).

Also, if we define

ψ ⁽ⁱ⁾ ₂ (x) = c ⁽ⁱ⁾ ₂

⁻¹

(x ² − a ⁽ⁱ⁾ ₁ x), α ⁽ⁱ⁾ ₂ = −c ⁽ⁱ⁾ ₂

⁻¹

v ⁽ⁱ⁾ , then

A n

_i−1

+2 = ψ ⁽ⁱ⁾ ₂ (A n

_i−1

+1 ) + α ⁽ⁱ⁾ ₂ (A 0 + · · · + A n

_i−1

).

Assume now that ψ ⁽ⁱ⁾ ₁ , . . . , ψ _i ⁽ⁱ⁾

₀

, α ⁽ⁱ⁾ ₁ , . . . , α ⁽ⁱ⁾ _i

₀

(2 ≤ i 0 < n i ) are defined so that

A n

_i−1

+j

0

= ψ ⁽ⁱ⁾ _j

₀

(A n

_i−1

+1 ) + α ⁽ⁱ⁾ _j

₀

(A 0 + · · · + A n

_i−1

), ψ ⁽ⁱ⁾ _j

₀

(0) = 0 holds for all 1 ≤ j 0 ≤ i 0 . Define the polynomial ψ _i ⁽ⁱ⁾

₀

₊₁ (x) of degree i 0 + 1 by:

ψ _i ⁽ⁱ⁾

₀

₊₁ (x) = c ⁽ⁱ⁾ _i

₀

₊₁

⁻¹

{xψ ⁽ⁱ⁾ _i

₀

(x) + α ⁽ⁱ⁾ _i

₀

(v 0 + · · · + v n

_i−1

)x

− b ⁽ⁱ⁾ _i

₀

₋₁ ψ ⁽ⁱ⁾ _i

₀

₋₁ (x) − a ⁽ⁱ⁾ _i

₀

ψ _i ⁽ⁱ⁾

₀

(x)}, and a constant α ⁽ⁱ⁾ _i

₀

₊₁ by:

α ⁽ⁱ⁾ _i

₀

₊₁ = −c ⁽ⁱ⁾ _i

₀

₊₁

⁻¹

(b ⁽ⁱ⁾ _i

₀

₋₁ α ⁽ⁱ⁾ _i

₀

₋₁ + a ⁽ⁱ⁾ _i

₀

α ⁽ⁱ⁾ _i

₀

).

Then

A n

_i−1

+i

0

+1 = ψ ⁽ⁱ⁾ _i

₀

₊₁ (A n

_i−1

+1 ) + α ⁽ⁱ⁾ _i

₀

₊₁ (A 0 + · · · + A n

_i−1

).

Observe here that ψ _i ⁽ⁱ⁾

₀

₊₁ (0) = 0.

(10)

(c) ⇒ (b) Here we must show the following: for i = 1, . . . , t, A n

_i−1

+1 A l = v l A n

_i−1

+1 (0 ≤ l ≤ n i−1 ), (7)

A n

_i−1

+1 A n

_i−1

+1 = v ⁽ⁱ⁾ (A 0 + · · · + A n

_i−1

) + a ⁽ⁱ⁾ ₁ A n

_i−1

+1 + c ⁽ⁱ⁾ ₂ A n

_i−1

+2

(8)

with c ⁽ⁱ⁾ ₂ 6= 0,

(9) A n

_i−1

+1 A n

_i−1

+i

0

= b ⁽ⁱ⁾ _i

₀

₋₁ A n

_i−1

+i

0

−1 + a ⁽ⁱ⁾ _i

₀

A n

_i−1

+i

0

+ c ⁽ⁱ⁾ _i

₀

₊₁ A n

_i−1

+i

0

+1 (2 ≤ i 0 ≤ n i ) with b ⁽ⁱ⁾ _i

₀

₋₁ 6= 0 (2 ≤ i 0 ≤ n i ), c ⁽ⁱ⁾ _i

₀

₊₁ 6= 0 (2 ≤ i 0 ≤ n i − 1), c ⁽ⁱ⁾ _n

_i

₊₁ = 0,

A n

_i−1

+1 A l = v ⁽ⁱ⁾ A l (n i + 1 ≤ l ≤ n).

(10)

By (4), (7), and (10) are true. Next, we show (8) and (9). For i = 1, we only need to show (9) for 1 ≤ l ≤ n 1 . By our assumption A i

0

= ψ _i ⁽¹⁾

₀

(A 1 ) for 0 ≤ i 0 ≤ n 1 . Since xψ ⁽¹⁾ _l (x) is a linear combination of ψ _l+1 ⁽¹⁾ (x), ψ ⁽¹⁾ _l (x), . . ., ψ ₀ ⁽¹⁾ (x), A 1 A l is a linear combination of A l+1 , A l , . . ., A 0 . Clearly, the coeffi- cient of A l+1 6= 0. As A 1 A l = P n

k=0 p ^k _1l A k , p ^k _1l = 0 for k ≥ l + 2 and p ^l+1 _1l 6= 0.

Since p ^k _1l = 0 ⇔ p ^l _1k = 0, p ^k _1l = 0 if |k − l| ≥ 2, and p ^k _1l 6= 0 if |k − l| = 1. Let i ≥ 2. First, we show (9). By (3) and (4), for 1 ≤ i 0 ≤ n i we have:

A n

_i−1

+i

0

A n

_i−1

+1 = ψ _i ⁽ⁱ⁾

₀

(A n

_i−1

+1 )A n

_i−1

+1 + α ⁽ⁱ⁾ _i

₀

(v 0 + · · · + v n

_i−1

)A n

_i−1

+1 . (11)

As ψ _k ⁽ⁱ⁾ (0) = 0 (1 ≤ k ≤ n i ),

ψ ⁽ⁱ⁾ _i

₀

(x)x + α ⁽ⁱ⁾ _i

₀

(v 0 + · · · + v n

_i−1

)x =

i

0

+1

X

k

0

=1

β k

0

ψ ⁽ⁱ⁾ _k

₀

(x) for some β k

0

with β i

0

+1 6= 0. Thus, for 2 ≤ i 0 ≤ n i ,

A n

_i−1

+i

0

A n

_i−1

+1 =

i

0

+1

X

k

0

=1

β k

0

A n

_i−1

+k

0

−

i

0

+1

X

k

0

=1

β k

0

α ⁽ⁱ⁾ _k

₀

(A 0 + · · · + A n

_i−1

),

by (3) and (11). Just as in the above argument, this implies that p ⁿ _n

ⁱ⁻¹

^+k

⁰

i−1

+1,n

_i−1

+l

0

= 0 for |k 0 − l 0 | ≥ 2, and p ⁿ _n

ⁱ⁻¹

^+k

⁰

i−1

+1,n

_i−1

+l

0

6= 0 for |k 0 − l 0 | = 1. Next, we show (8). By (3) with i 0 = 2,

A n

_i−1

+2 = aA ² _n

_i−1

₊₁ + bA n

_i−1

+1 + α ⁽ⁱ⁾ ₂ (A 0 + · · · + A n

_i−1

) for some a, b with a 6= 0. So

A n

_i−1

+1 A n

_i−1

+1 = w(A 0 + · · · + A n

_i−1

) + a ⁽ⁱ⁾ ₁ A n

_i−1

+1 + c ⁽ⁱ⁾ ₂ A n

_i−1

+2

(11)

with c ⁽ⁱ⁾ ₂ 6= 0, where we put

w = −a ⁻¹ α ⁽ⁱ⁾ ₂ , a ⁽ⁱ⁾ ₁ = −a ⁻¹ b, c ⁽ⁱ⁾ ₂ = a ⁻¹ .

It remains to see that w = v ⁽ⁱ⁾ . For 0 ≤ k ≤ n i−1 , let (x, y) ∈ R k . Then x i = y i , . . . , x t = y t . Thus p ^k _n

i−1

+1,n

_i−1

+1 = v ⁽ⁱ⁾ , and hence w = v ⁽ⁱ⁾ . (c) ⇔ (d) This is straightforward.

(a) ⇔ (e) Note that (2) is equivalent to: for i = 1, . . . , t,

(x i , y i ) ∈ R ⁽ⁱ⁾ _j ⇔ ∂ i (x i , y i ) = j (0 ≤ j ≤ n i ) (cf. [2, Prop. 1.1, p. 189]). Remark 3. (1) When X is a weak metric scheme, for i = 1, . . . , t the submatrix [L ⁽ⁱ⁾ ] of L ⁽ⁱ⁾ will be denoted by:

[L ⁽ⁱ⁾ ] =



 



a ⁽ⁱ⁾ ₀ b ⁽ⁱ⁾ ₀ c ⁽ⁱ⁾ ₁ a ⁽ⁱ⁾ ₁ b ⁽ⁱ⁾ ₁

. .. . .. . ..

c ⁽ⁱ⁾ _n

_i

₋₁ a ⁽ⁱ⁾ _n

_i

₋₁ b ⁽ⁱ⁾ _n

_i

₋₁ c ⁽ⁱ⁾ n

i

a ⁽ⁱ⁾ n

i



 

 . (12)

Observe here that a ⁽ⁱ⁾ ₀ = 0, b ⁽ⁱ⁾ ₀ = v ⁽ⁱ⁾ , c ⁽ⁱ⁾ ₁ = v n

_i−1

. (2) Put

ψ ⁽¹⁾ (x) = ψ ⁽¹⁾ ₀ (x) + · · · + ψ n ⁽¹⁾

1

(x), ψ ⁽ⁱ⁾ (x) = ψ ⁽ⁱ⁾ ₁ (x) + · · · + ψ n ⁽ⁱ⁾

_i

(x) (2 ≤ i ≤ t).

Then ψ ⁽ⁱ⁾ (x) is a polynomial of degree n i for 1 ≤ i ≤ t. Using (3), one can show by induction on i that, for i = 2, . . . , t,

A 0 + · · · + A n

_i−1

= X i−1 k=1

c ⁽ⁱ⁾ _k ψ ^(k) (A n

_k−1

+1 ), (13)

where c ⁽ⁱ⁾ _k = Q i−1

l=k+1 (1 + α ^(l) ₁ + · · · + α ^(l) n

_l

) with the understanding that c ⁽ⁱ⁾ _i−1 = 1.

(3) For i = 2, . . . , t, 1 ≤ i 0 ≤ n i ,

A n

_i−1

+i

0

= ψ _i ⁽ⁱ⁾

₀

(A n

_i−1

+1 ) + α ⁽ⁱ⁾ _i

₀

ξ ⁽ⁱ⁾ _n

_i−1

₊₁ (A n

_i−2

+1 ),

where ξ ⁽ⁱ⁾ n

_i−1

+1 (x) is a polynomial with real coefficients of degree n i−1 + 1.

Indeed, from (3) and (13), we have

A n

_i−1

+i

0

= ψ ⁽ⁱ⁾ _i

₀

(A n

_i−1

+1 ) + α ⁽ⁱ⁾ _i

₀

X i−1 k=1

c ⁽ⁱ⁾ _k ψ ^(k) (A n

_k−1

+1 ).

(14)

Multiplying both sides of (14) by A n

_i−2

+1 and using (4), we get:

v ⁽ⁱ⁻¹⁾ A n

_i−1

+i

0

= v ⁽ⁱ⁻¹⁾ ψ _i ⁽ⁱ⁾

₀

(A n

_i−1

+1 )

(12)

+ α ⁽ⁱ⁾ _i

₀

X i−2 k=1

c ⁽ⁱ⁾ _k ψ ^(k) (A n

_k−1

+1 )A n

_i−2

+1 + α ⁽ⁱ⁾ _i

₀

ψ ⁽ⁱ⁻¹⁾ (A n

_i−2

+1 )A n

_i−2

+1

= v ⁽ⁱ⁻¹⁾ ψ _i ⁽ⁱ⁾

₀

(A n

_i−1

+1 ) + α ⁽ⁱ⁾ _i

₀

{

X i−2 k=1

c ⁽ⁱ⁾ _k ψ ^(k) (v ^(k) )A n

_i−2

+1 + ψ ⁽ⁱ⁻¹⁾ (A n

_i−2

+1 )A n

_i−2

+1 }.

This implies

A n

_i−1

+i

0

= ψ _i ⁽ⁱ⁾

₀

(A n

_i−1

+1 ) + α ⁽ⁱ⁾ _i

₀

ξ ⁽ⁱ⁾ n

_i−1

+1 (A n

_i−2

+1 ), where we put

ξ _n ⁽ⁱ⁾

i−1

+1 (x) = v ⁽ⁱ⁻¹⁾

⁻¹

( _i−2

X

k=1

c ⁽ⁱ⁾ _k ψ ^(k) (v ^(k) )x + ψ ⁽ⁱ⁻¹⁾ (x)x )

.

Proposition 4. Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a weak metric scheme with X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

). Let the entries of the submatrix [L ⁽ⁱ⁾ ] of L ⁽ⁱ⁾ be as in (12). Then

(a)

(15) a ⁽¹⁾ _l + b ⁽¹⁾ _l + c ⁽¹⁾ _l = v ⁽¹⁾ for i ≥ 2, (16) a ⁽ⁱ⁾ _l + b ⁽ⁱ⁾ _l + c ⁽ⁱ⁾ _l = v ⁽ⁱ⁾ , 0 ≤ l ≤ n i and l 6= 1,

v ⁽ⁱ⁾ − (v 0 + v 1 + · · · + v n

_i−1

−1 ), l = 1.

Here c ⁽ⁱ⁾ ₀ = 0, b ⁽ⁱ⁾ n

_i

= 0 for i = 1, . . . , t.

(b) v n

_i−1

+i

0

= v ⁽ⁱ⁾ b ⁽ⁱ⁾ ₁ b ⁽ⁱ⁾ ₂ · · · b ⁽ⁱ⁾ _i

₀

₋₁ /c ⁽ⁱ⁾ ₂ c ⁽ⁱ⁾ ₃ · · · c ⁽ⁱ⁾ _i

₀

for i = 1, . . . , t, 2 ≤ i 0 ≤ n i . (c) v ⁽ⁱ⁾ = b ⁽ⁱ⁾ ₀ ≥ b ⁽ⁱ⁾ ₁ ≥ · · · ≥ b ⁽ⁱ⁾ _n

_i

₋₁ for i = 1, . . . , t.

(d) v n

_i−1

= c ⁽ⁱ⁾ ₁ ≤ c ⁽ⁱ⁾ ₂ ≤ · · · ≤ c ⁽ⁱ⁾ n

_i

for i = 1, . . . , t.

Proof. (a) The result follows from P n j=0 p ^k _n

i−1

+1,j = v ⁽ⁱ⁾ , Theorem 2(b)(i), (iv), (v), and (12).

(b) Since p ^k _ij v k = p ^j _ik v j , v n

_i−1

+i

0

b ⁽ⁱ⁾ _i

₀

= v n

_i−1

+i

0

+1 c ⁽ⁱ⁾ _i

₀

₊₁ . So

v ⁽ⁱ⁾ = c ⁽ⁱ⁾ ₂ v n

_i−1

+2

b ⁽ⁱ⁾ ₁ = c ⁽ⁱ⁾ ₂ c ⁽ⁱ⁾ ₃ v n

_i−1

+3

b ⁽ⁱ⁾ ₁ b ⁽ⁱ⁾ ₂ = · · · = c ⁽ⁱ⁾ ₂ · · · c ⁽ⁱ⁾ _i

₀

v n

_i−1

+i

0

b ⁽ⁱ⁾ ₁ b ⁽ⁱ⁾ ₂ · · · b ⁽ⁱ⁾ _i

₀

₋₁ . (c) Let x, y ∈ X with d(x, y) = n i−1 + i 0 (1 ≤ i 0 ≤ n i − 1). Let w be an element of X such that d(x, w) = n i−1 + 1, d(w, y) = n i−1 + i 0 + 1. As

∂ i (x i , y i ) = i 0 , we can choose z i ∈ X i , with ∂ i (x i , z i ) = i 0 − 1, ∂ i (z i , y i ) = 1.

Then ∂ i (z i , w i ) ≤ i 0 . Also, ∂ i (z i , w i ) ≥ i 0 , since otherwise we would have

∂ i (w i , y i ) ≤ i 0 . We put z = (x 1 , . . . , x i−2 , z i−1 , z i , x i+1 , . . . , x t ), where z i−1 ∈

X i−1 is an element with ∂ i−1 (x i−1 , z i−1 ) = n i−1 . Note here that such an

z i−1 can be chosen, since, if necessary, after replacing (i − 1)th component

(13)

x i−1 of x we may assume {a i−1 ∈ X i−1 | ∂ i−1 (x i−1 , a i−1 ) = n i−1 } 6= ∅.

Then d(x, z) = n i−1 + i 0 − 1. Observe that this holds even for i 0 = 1. Also, d(w, z) = n i−1 + i 0 . This shows that b ⁽ⁱ⁾ _i

₀

≤ b ⁽ⁱ⁾ _i

₀

₋₁ for 1 ≤ i 0 ≤ n i − 1.

(d) Let x, z ∈ X, with d(x, z) = n i−1 + i 0 + 1 (1 ≤ i 0 ≤ n i − 1). Choose y i ∈ X i with ∂ i (x i , y i ) = i 0 , ∂ i (y i , z i ) = 1. Put y = (x 1 , . . . , x i−1 , y i , x i+1 , . . . , x t ).

Then d(x, y) = n i−1 +i 0 . Let w be an element of X such that d(x, w) = n i−1 +1, d(w, y) = n i−1 + i 0 − 1. Just as in the proof of (c), we have ∂ i (w i , z i ) = i 0 , and hence d(w, z) = n i−1 + i 0 . This shows c ⁽ⁱ⁾ _i

₀

≤ c ⁽ⁱ⁾ _i

₀

₊₁ for 1 ≤ i 0 ≤ n i − 1.

4. Parameters of weak metric schemes

Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a symmetric association scheme which is given as the wreath product of the symmetric association schemes X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

). Then we will determine some of the parameters of X, such as valencies, multiplicities, the adjacency matrices, the irreducible idempotents, P -matrix and Q-matrix. Further, we will determine the first intersection matrix for each level and the first dual intersection matrix for each level. Also, we will prove Theorem 11 which is an analogue of Theorem 1.3 in [2, p. 197].

In below, all the matrices in the i-th factor of tensor products are of size |X i | for i = 1, . . . , t, and accordingly I and J will denote respectively the identity and all-one matrices of various sizes.

Theorem 5. Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a symmetric association scheme which is given as the wreath product of the symmetric association schemes X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

). Then we have the following.

(a) |X 1 | · · · |X i−1 | v _i ⁽ⁱ⁾

₀

= v n

_i−1

+i

0

for 1 ≤ i ≤ t, 1 ≤ i 0 ≤ n i , or i = 1, i 0 = 0.

(b) λ i := P ⁿ

i

j=0 v j = |X 1 | · · · |X i | for i = 1, . . . , t.

(c) |X t | · · · |X i+1 | m ⁽ⁱ⁾ _i

₀

= m n

_i+1

+i

0

for 1 ≤ i ≤ t, 1 ≤ i 0 ≤ n i , or i = t, i 0 = 0.

(d) µ i := P n

i

j=0 m j = |X t | · · · |X i | for i = 1, . . . , t.

(e)

(17)

A 0 = A ⁽¹⁾ ₀ ⊗ I ⊗ · · · ⊗ I = I, A 1 = A ⁽¹⁾ ₁ ⊗ I ⊗ · · · ⊗ I, . . . , A n

1

= A ⁽¹⁾ _n

₁

⊗ I ⊗ · · · ⊗ I, A n

1

+1 = J ⊗ A ⁽²⁾ ₁ ⊗ I ⊗ · · · ⊗ I, . . . , A n

1

+n

2

= J ⊗ A ⁽²⁾ _n

₂

⊗ I ⊗ · · · ⊗ I, . . . ,

A n

_i−1

+i

0

= J ⊗ · · · ⊗ J ⊗ A ⁽ⁱ⁾ _i

₀

⊗ I ⊗ · · · ⊗ I (1 ≤ i 0 ≤ n i ), . . . ,

A n

_t−1

+1 = J ⊗ · · · ⊗ J ⊗ A ^(t) ₁ , . . . , A n = J ⊗ · · · ⊗ J ⊗ A ^(t) _n

_t

,

where in (17) there are (i − 1) factors of J.

(14)

(f)

E 0 = E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ^(t−1) ⊗ E ₀ ^(t) = |X| ⁻¹ J, E 1 = E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ^(t−1) ⊗ E ₁ ^(t) ,

.. .

E n

t

= E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ^(t−1) ⊗ E _n ^(t)

_t

, E n

t

+1 = E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ^(t−2) ⊗ E ₁ ^(t−1) ⊗ I,

.. .

E n

t

+n

_t−1

= E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ^(t−2) ⊗ E _n ^(t−1)

_t−1

⊗ I, .. .

E n

i+1

+i

0

= E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E _i ⁽ⁱ⁾

₀

⊗ I ⊗ · · · ⊗ I (1 ≤ i 0 ≤ n i ) .. .

E n

2

+1 = E ₁ ⁽¹⁾ ⊗ I ⊗ · · · ⊗ I, . . . , E n = E n ⁽¹⁾

1

⊗ I ⊗ · · · ⊗ I.

For i = 1, 2, . . . , t, let P ⁽ⁱ⁾ and Q ⁽ⁱ⁾ be respectively the P -matrix and the Q- matrix of X ⁽ⁱ⁾ . For these, we put

P ⁽ⁱ⁾ =



 

 

1 v ₁ ⁽ⁱ⁾ · · · v ⁽ⁱ⁾ n

_i

1 .. . P ˜ ⁽ⁱ⁾ 1



 

 

, Q ⁽ⁱ⁾ =



 

 

1 m ⁽ⁱ⁾ ₁ · · · m ⁽ⁱ⁾ n

_i

1 .. . Q ˜ ⁽ⁱ⁾ 1



 

  .

(g) P =



 



1 v 1 · · · v n

1

v n

1

+1 · · · v n

2

· · · · v n

t−2

+1 · · · v n

t−1

v n

t−1

+1 · · · v n

1 v 1 v n

1

v n

1

+1 v n

2

· · · · v n

t−2

+1 v n

t−1

.. . .. . .. . .. . .. . .. . .. . λ _t−1 P ˜ ^(t)

1 v 1 v n

1

v n

1

+1 v n

2

· · · · v n

t−2

+1 v n

t−1

1 v 1 v n

1

v n

1

+1 v n

2

· · · ·

.. . .. . .. . .. . .. . λ _t−2 P ˜ ^(t−1) 0

1 v ₁ v n

1

v n

1

+1 v n

2

· · · ·

.. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

1 v ₁ v n

1

· · · ·

.. . .. . .. . λ ₁ P ˜ ⁽²⁾ 0 0

1 v ₁ v n

1

· · · ·

1 · · · ·

.. . λ ₀ P ˜ ⁽¹⁾ 0 0 0

1 · · · ·



 



.

(15)

(h) Q =



 



1 m 1 · · · m n

t

m n

t

+1 · · · m n

t−1

· · · · m n

3

+1 · · · m n

2

m n

2

+1 · · · m n

1

1 m 1 m n

t

m n

t

+1 m n

t−1

· · · · m n

3

+1 m n

2

.. . .. . .. . .. . .. . .. . .. . µ 2 Q ˜ ⁽¹⁾

1 m 1 m n

t

m n

t

+1 m n

t−1

· · · · m n

3

+1 m n

2

1 m 1 m n

t

m n

t

+1 m n

t−1

· · · ·

.. . .. . .. . .. . .. . µ 3 Q ˜ ⁽²⁾ 0

1 m 1 m n

t

m n

t

+1 m n

t−1

· · · ·

.. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

1 m 1 m n

t

· · · ·

.. . .. . .. . µ t Q ˜ ^(t−1) 0 0

1 m 1 m n

t

· · · ·

1 · · · ·

.. . µ t+1 Q ˜ ^(t) 0 0 0

1 · · · ·



 

 .

Proof. (a) Let (x 1 , x 2 , . . . , x t ) ∈ X. Then

v n

_i−1

+i

0

=| {(y 1 , . . . , y i , x i+1 , . . . , x t ) ∈ X | (x i , y i ) ∈ R _i ⁽ⁱ⁾

₀

}|

=|X 1 | · · · |X i−1 |v ⁽ⁱ⁾ _i

₀

. (b) Induction on i. For i = 1, λ 1 = P n

1

j=0 v j = P n

1

j=0 v ⁽¹⁾ _j = |X 1 |, by (a).

Assume that it holds for i (i ≥ 1). Then, again by (a),

λ i+1 = λ i +

n

i+1

X

j=n

i

+1

v j

= |X 1 | · · · |X i | + |X 1 | · · · |X i |

n

_i+1

X

i

0

=1

v ⁽ⁱ⁺¹⁾ _i

₀

= |X 1 | · · · |X i | + |X 1 | · · · |X i |(|X i+1 | − 1)

= |X 1 | · · · |X i+1 |.

(e), (f) These can be shown directly. Or these will follow by generalizing the results in Section 4 of [9] stated for the wreath product of two association schemes. Observe, however, that in what will follow we reversed the order of factors in tensor product expressions of adjacency matrices and irreducible idempotents.

(c) From (f),

m n

_i+1

+i

0

= trE n

_i+1

+i

0

= trE ₀ ⁽¹⁾ · · · trE ₀ ⁽ⁱ⁻¹⁾ trE _i ⁽ⁱ⁾

₀

|X i+1 | · · · |X t |

= |X t | · · · |X i+1 | m ⁽ⁱ⁾ _i

₀

.

(d) Reverse induction on i, starting from t.

(16)

(g) For the wreath product of two association schemes, this is also stated in Section 4 of [9]. The proof is left to the reader, as it can be shown just as in (h) below.

(h)

E n

i+1

+i

0

= E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E _i ⁽ⁱ⁾

₀

⊗ I ⊗ · · · ⊗ I

= |X 1 | ⁻¹ J ⊗ · · · ⊗ |X i−1 | ⁻¹ J ⊗ |X i | ⁻¹

n

_i

X

j=0

q ⁽ⁱ⁾ _ji

₀

A ⁽ⁱ⁾ _j ⊗ I ⊗ · · · ⊗ I

= |X| ⁻¹ {µ i+1 n

_i

X

j=1

q _ji ⁽ⁱ⁾

₀

J ⊗ · · · ⊗ J ⊗ A ⁽ⁱ⁾ _j ⊗ I ⊗ · · · ⊗ I + m n

i+1

+i

0

J ⊗ · · · ⊗ J ⊗ I ⊗ I ⊗ · · · ⊗ I}

= |X| ⁻¹ {µ i+1 n

i

X

j=1

q _ji ⁽ⁱ⁾

₀

A n

_i−1

+j + m n

i+1

+i

0

(A 0 + A 1 + · · · + A n

_i−1

)}, where the last equality is obtained by replacing various J by the corresponding sum of adjacency matrices for X ^(j) for j = i − 1, i − 2, . . . , 1. Remark 6. From the expression of P in Theorem 5 (g), we have:

θ ₀ ⁽ⁱ⁾ , θ ⁽ⁱ⁾ ₁ , . . . , θ ⁽ⁱ⁾ _n = θ ⁽ⁱ⁾ ₀ , θ ⁽ⁱ⁾ ₀ , . . . , θ ⁽ⁱ⁾ ₀

| {z }

n

_i+1

+1

, θ ⁽ⁱ⁾ _n

_i+1

₊₁ , . . . , θ ⁽ⁱ⁾ _n

_i

, 0, . . . , 0

| {z }

n

_i−1

.

Theorem 7. Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a symmetric association scheme which is given as the wreath product of the symmetric association schemes X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

).

(a) The first intersection matrix L ⁽ⁱ⁾ = (p _n ^k

_i−1

_{+1, l} ) 0≤k,l≤n of level i is

L ⁽ⁱ⁾ =



 



0 v ⁽ⁱ⁾ .. . v ⁽ⁱ⁾

0 v 0 v 1 · · · v n

_i−1

λ i−1 L ˜ ⁽ⁱ⁾ ₁

0 v ⁽ⁱ⁾ I n−n

i



 

 , (18)

where the upper left diagonal block matrix in (18) is the zero matrix of size n i−1 + 1 (with rows and columns indexed by 0, 1, . . . , n i−1 ), and ˜ L 1

(i) is the submatrix (p ^(i)k _1l ) 1≤k,l≤n

_i

of the first intersection matrix L ⁽ⁱ⁾ ₁ = (p ^(i)k _1l ) 0≤k,l≤n

_i

of X ⁽ⁱ⁾ .

(17)

In particular, the submatrix [L ⁽ⁱ⁾ ] = (p ^k _n

_i−1

_+1,l ) n

_i−1

≤k,l≤n

i

of L ⁽ⁱ⁾ is

[L ⁽ⁱ⁾ ] =



 



0 v ⁽ⁱ⁾ 0 · · · 0 v n

_i−1

0 λ i−1 L ˜ 1 (i)

.. . 0



 

 . (19)

(b) The first dual intersection matrix M ⁽ⁱ⁾ = (q ^k _n

_i+1

_+1,l ) 0≤k,l≤n of level i is

M ⁽ⁱ⁾ =



 



0 m ⁽ⁱ⁾ .. . m ⁽ⁱ⁾

0 m 0 m 1 · · · m n

i+1

µ i+1 M ˜ ₁ ⁽ⁱ⁾

0 m ⁽ⁱ⁾ I n−n

i



 

 , (20)

where the upper left diagonal block matrix in (20) is the zero matrix of size n i+1 + 1 (with rows and columns indexed by 0, 1, . . . , n i+1 ), and ˜ M ₁ ⁽ⁱ⁾ is the submatrix (q ^(i)k _1l ) 1≤k,l≤n

i

of the first intersection matrix M ₁ ⁽ⁱ⁾ = (q ^(i)k _1l ) 0≤k,l≤n

i

of X ⁽ⁱ⁾ .

In particular, the submatrix [M ⁽ⁱ⁾ ] = (q ^k _n

_i+1

_+1,l ) n

_i+1

≤k,l≤n

i

of M ⁽ⁱ⁾ is

[M ⁽ⁱ⁾ ] =



 



0 m ⁽ⁱ⁾ 0 · · · 0 m n

_i+1

0 µ i+1 M ˜ 1 (i)

.. . 0



 

 . (21)

Proof. Here we will give only a proof for (b). From Theorem 5(f), E n

_i+1

+1 ◦ E n

_i+1

+l

= (E ⁽¹⁾ ₀ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E ₁ ⁽ⁱ⁾ ⊗ I ⊗ · · · ⊗ I)

◦ (E ⁽¹⁾ ₀ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E _l ⁽ⁱ⁾ ⊗ I ⊗ · · · ⊗ I)

= (|X 1 | · · · |X i−1 |) ⁻¹ E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E ₁ ⁽ⁱ⁾ ◦ E _l ⁽ⁱ⁾ ⊗ I ⊗ · · · ⊗ I

= |X| ⁻¹ {

n

_i

X

k=1

µ i+1 q _1l ^(i)k E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E _k ⁽ⁱ⁾ ⊗ I ⊗ · · · ⊗ I

+ m ⁽ⁱ⁾ δ 1l E ⁽¹⁾ ₀ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E ₀ ⁽ⁱ⁾ ⊗ I ⊗ · · · ⊗ I}

(18)

= |X| ⁻¹ ( _n

_i

X

k=1

µ i+1 q _1l ^(i)k E n

i+1

+k + m ⁽ⁱ⁾ δ 1l (E 0 + · · · + E n

i+1

) )

,

where the last equality is obtained by replacing various I by the corresponding sum of irreducible idempotents for X ^(j) , for j = i + 1, . . . , t.

Let j > i, 1 ≤ l ≤ n j , or let j = t, l = 0. Then E n

_i+1

+1 ◦ E n

_j+1

+l

= (E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E ₁ ⁽ⁱ⁾ ⊗ I ⊗ · · · ⊗ I ⊗ I ⊗ I ⊗ · · · ⊗ I)

◦ (E ₀ ⁽¹⁾ ⊗ · · · ⊗ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E ₀ ⁽ⁱ⁾ ⊗ E ₀ ⁽ⁱ⁺¹⁾ ⊗ · · · ⊗ E ^(j−1) ₀ ⊗ E ^(j) _l ⊗ I ⊗ · · · ⊗ I)

= |X 1 | ⁻¹ E ₀ ⁽¹⁾ ⊗ · · · ⊗ |X i−1 | ⁻¹ E ₀ ⁽ⁱ⁻¹⁾ ⊗ E ₁ ⁽ⁱ⁾ ◦ E ₀ ⁽ⁱ⁾ ⊗ |X i+1 | ⁻¹ I ⊗ · · ·

⊗ |X j−1 | ⁻¹ I ⊗ I ◦ E _l ^(j) ⊗ I ⊗ · · · ⊗ I

= |X| ⁻¹ m n

j+1

+l E n

i+1

+1 , where we note that

E ₁ ⁽ⁱ⁾ ◦ E ₀ ⁽ⁱ⁾ = |X i | ⁻¹ E ₁ ⁽ⁱ⁾ , I ◦ E _l ^(j) = |X j | ⁻¹ m ^(j) _l I.

Similarly to the just above case, one shows, for j < i, that E n

i+1

+1 ◦

E n

_j+1

+l = |X| ⁻¹ m ⁽ⁱ⁾ E n

_j+1

+l .

Remark 8. λ i−1 L ⁽ⁱ⁾ ₁ equals [L ⁽ⁱ⁾ ] except for the (2,1) entry, namely the (2,1) entry of λ i−1 L ⁽ⁱ⁾ ₁ is λ i−1 = P ⁿ

i−1

j=0 v j , whereas that of [L ⁽ⁱ⁾ ] is v n

_i−1

(cf. (19)).

Also, µ i+1 M ₁ ⁽ⁱ⁾ equals [M ⁽ⁱ⁾ ] except for the (2,1) entry, namely the (2,1) entry of µ i+1 M ₁ ⁽ⁱ⁾ is µ i+1 = P n

i+1

j=0 m j , whereas that of [M ⁽ⁱ⁾ ] is m n

i+1

(cf. (21)).

Proposition 9. Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a symmetric association scheme which is given as the wreath product of the symmetric association schemes X ⁽ⁱ⁾ = (X i , {R ⁽ⁱ⁾ _j } ⁿ _j=0

ⁱ

). Then we have:

(a) θ ⁽ⁱ⁾ ₀ , . . . , θ n ⁽ⁱ⁾ are real numbers and eigenvalues of L ⁽ⁱ⁾ for i = 1, 2, . . . , t.

Further, |θ _j ⁽ⁱ⁾ | ≤ v ⁽ⁱ⁾ for i = 1, 2, . . . , t, 0 ≤ j ≤ n.

If in addition X is a weak metric scheme, then we have:

(b)

(22)

ψ ₀ ⁽¹⁾ (x) = 1, ψ ₁ ⁽¹⁾ (x) = x,

c ⁽¹⁾ _i

₀

ψ _i ⁽¹⁾

₀

(x) = (x − a ⁽¹⁾ _i

₀

₋₁ )ψ _i ⁽¹⁾

₀

₋₁ (x) − b ⁽¹⁾ _i

₀

₋₂ ψ ⁽¹⁾ _i

₀

₋₂ (x) (2 ≤ i 0 ≤ n 1 ) α ⁽¹⁾ ₀ = α ⁽¹⁾ ₁ = · · · = α ⁽¹⁾ _n

₁

= 0 for 2 ≤ i ≤ t,

(23)

ψ ₁ ⁽ⁱ⁾ (x) = x, ψ ⁽ⁱ⁾ ₂ (x) = c ⁽ⁱ⁾⁻¹ ₂ (x ² − a ⁽ⁱ⁾ ₁ x), c ⁽ⁱ⁾ _i

₀

ψ _i ⁽ⁱ⁾

₀

(x) = (x − a ⁽ⁱ⁾ _i

₀

₋₁ )ψ _i ⁽ⁱ⁾

₀

₋₁ (x) − b ⁽ⁱ⁾ _i

₀

₋₂ ψ _i ⁽ⁱ⁾

₀

₋₂ (x)

+ α ⁽ⁱ⁾ _i

₀

₋₁ (v 0 + · · · + v n

_i−1

)x (3 ≤ i 0 ≤ n i ),

(19)

(24) α ⁽ⁱ⁾ ₁ = 0, α ⁽ⁱ⁾ ₂ = −c ⁽ⁱ⁾ ₂

⁻¹

v ⁽ⁱ⁾ = −c ⁽ⁱ⁾ ₂

⁻¹

b ⁽ⁱ⁾ ₀ ,

c ⁽ⁱ⁾ _i

₀

α ⁽ⁱ⁾ _i

₀

= −(b ⁽ⁱ⁾ _i

₀

₋₂ α ⁽ⁱ⁾ _i

₀

₋₂ + a ⁽ⁱ⁾ _i

₀

₋₁ α ⁽ⁱ⁾ _i

₀

₋₁ ) (3 ≤ i 0 ≤ n i ).

Proof. (a) The first statement follows from P L ⁽ⁱ⁾ P ⁻¹ = diag(θ ⁽ⁱ⁾ ₀ , . . . , θ ⁽ⁱ⁾ n ) and the second one does from Perron-Frobenius’ theorem.

(b) This follows from the proof (b) ⇒ (c) of Theorem 2. Lemma 10. Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a weak metric scheme. Then, for 2 ≤ i ≤ t,

(1 + α ⁽ⁱ⁾ ₂ + · · · + α ⁽ⁱ⁾ _n

_i

)λ i−1 = (−1) ⁿ

ⁱ

θ ⁽ⁱ⁾ _n

_i+1

₊₁ · · · θ ⁽ⁱ⁾ _n

_i

c ⁽ⁱ⁾ ₂ c ⁽ⁱ⁾ ₃ · · · c ⁽ⁱ⁾ n

i

. (25)

Proof. From (24) and in view of Remark 8, with α ⁽ⁱ⁾ ₀ = 1 we have:



 

 1

˜b ⁽ⁱ⁾ ₀ ˜ c ⁽ⁱ⁾ ₂

˜

a ⁽ⁱ⁾ ₂ ˜ c ⁽ⁱ⁾ ₃

˜b ⁽ⁱ⁾ ₂ ˜ a ⁽ⁱ⁾ ₃ c ˜ ⁽ⁱ⁾ ₄ . .. . .. . ..

˜b ⁽ⁱ⁾ n

_i

−2 ˜ a ⁽ⁱ⁾ n

_i

−1 ˜ c ⁽ⁱ⁾ n

_i



 





 

  α ⁽ⁱ⁾ ₀ α ⁽ⁱ⁾ ₂ α ⁽ⁱ⁾ ₃ .. . .. . α ⁽ⁱ⁾ n

_i



 

 

=



 

 1 0 0 .. . .. . 0



 

 (26)

if the first intersection matrix L ⁽ⁱ⁾ ₁ of X ⁽ⁱ⁾ is given by



 



˜ a ⁽ⁱ⁾ ₀ ˜b ⁽ⁱ⁾ ₀

˜

c ⁽ⁱ⁾ ₁ ˜ a ⁽ⁱ⁾ ₁ ˜b ⁽ⁱ⁾ ₁

˜

c ⁽ⁱ⁾ ₂ ˜ a ⁽ⁱ⁾ ₂ ˜b ⁽ⁱ⁾ ₂ . .. . .. . ..

˜

c ⁽ⁱ⁾ _n

_i

₋₁ ˜ a ⁽ⁱ⁾ _n

_i

₋₁ ˜b ⁽ⁱ⁾ _n

_i

₋₁

˜

c ⁽ⁱ⁾ n

i

˜ a ⁽ⁱ⁾ n

i



 

 .

By Remark 8 and the shape of P (cf. Theorem 5(g)), (25) is equivalent to:

(1 + α ⁽ⁱ⁾ ₂ + · · · + α ⁽ⁱ⁾ n

_i

) = (−1) ⁿ

ⁱ

p ⁽ⁱ⁾ ₁₁ · · · p ⁽ⁱ⁾ _n

_i

₁

˜

c ⁽ⁱ⁾ ₂ ˜ c ⁽ⁱ⁾ ₃ · · · ˜ c ⁽ⁱ⁾ n

_i

, (27)

with P ⁽ⁱ⁾ = (p ⁽ⁱ⁾ _kl ) 0≤k, l≤n

i

the P -matrix of X ⁽ⁱ⁾ .

It can be shown (cf. [2, Theorem 1.3, p. 197]) that, if we define

˜

c ⁽ⁱ⁾ _j P j (x) = (x − ˜ a ⁽ⁱ⁾ _j−1 )P j−1 (x) − ˜b ⁽ⁱ⁾ _j−2 P j−2 (x) (2 ≤ j ≤ n i ), P 0 (x) = 1, P 1 (x) = x,

and

P n

_i

+1 (x) = (x − ˜ a ⁽ⁱ⁾ _n

_i

)P n

_i

(x) − ˜b ⁽ⁱ⁾ _n

_i

₋₁ P n

_i

−1 (x),

(20)

then

P n

i

+1 (x) = 1

˜

c ⁽ⁱ⁾ ₂ c ˜ ⁽ⁱ⁾ ₃ · · · ˜ c ⁽ⁱ⁾ n

_i

(x − p ⁽ⁱ⁾ ₀₁ )(x − p ⁽ⁱ⁾ ₁₁ ) · · · (x − p ⁽ⁱ⁾ _n

_i

₁ )

= (x − p ⁽ⁱ⁾ ₀₁ )

n

_i

X

j=0

P j (x).

So we see that P n

i

j=0 P j (0) = RHS of (27), and (26) is satisfied with α ⁽ⁱ⁾ ₀ , α ⁽ⁱ⁾ ₂ , . . ., α ⁽ⁱ⁾ n

i

respectively replaced by P 0 (0), P 2 (0), . . . , P n

_i

(0). As X ⁽ⁱ⁾ is a metric scheme, the square matrix on the LHS of (26) is invertible, and hence α ⁽ⁱ⁾ ₀ = P 0 (0), α ⁽ⁱ⁾ ₂ = P 2 (0), . . . , α ⁽ⁱ⁾ n

i

= P n

_i

(0). In particular, P n

_i

j=0 P j (0) =

α ⁽ⁱ⁾ ₀ + α ⁽ⁱ⁾ ₂ + · · · + α ⁽ⁱ⁾ n

_i

.

Theorem 11. Let X = (X = X 1 × · · · × X t , {R j } ⁿ _j=0 ) = X ⁽¹⁾ ≀ · · · ≀ X ^(t) be a weak metric scheme. Assume ψ ⁽ⁱ⁾ (x)’s are as in (22) and (23). Let ψ _n ⁽¹⁾

Weak metric and weak cometric schemes

J. Korean Math. Soc. 46 (2009), No. 4, pp. 785–812 DOI 10.4134/JKMS.2009.46.4.785