MODULE CORRESPONDENCES IN SOME BLOCKS OF FINITE GENERAL LINEAR GROUPS

AKIHIKO HIDA AND HYOHE MIYACHI

1. Introduction

Let ℓ be a prime number. Let O be a complete discrete valuation ring with fraction field K of characteristic 0 and residue field k = O/J(O) of characteristic ℓ. We assume that K is big enough for all groups in this paper. Let GL_{n}(q) be the general linear group over field of q elements, where q is a prime power. We assume that ℓ dose not divide q. Let e be the multiplicative order of q in 𝕜. The unipotent blocks of OGL_{n}(q) are parametrized by e-weights (semisimple parts) and e-cores (unipotent parts) [7]. Let B_{w,ρ} be the unipotent block of OGL_{n}(q) with e-weight w and e-core ρ.

Let ρ be an e-core satisfying the following property : ρ has an (e-runner) abacus representation such that Γ_{i−1}+w−1 ≤ Γ_{i} where Γ_{i} is the number of beads on the ith runner.[18],[4]. Let m = ew+|ρ|. Let N_{w} = (GL_{e}(q)≀𝔖_{w})×GL_{|ρ|}(q). Let f be the block idempotent of B_{0}(O(GL_{e}(q) ≀ 𝔖_{w})) ⊗ B_{0,ρ} where B_{0}(O(GL_{e}(q) ≀ 𝔖_{w})) is the principal block of the wreath product GL_{e}(q) ≀ 𝔖_{w}. Let D be a Sylow ℓsubgroup of GL_{e}(q) ≀ 𝔖_{w}. Then, as an O(GL_{m}(q) × N_{w})-module, B_{w,ρ}f has a unique indecomposable direct summand X_{w} with vertex ΔD = {(d, d) | d ∈ D}. W.Turner [19] proved the following which is analogoues to the result of Chuang and Kesser for symmetric groups [4].

Theorem 1.1. If w < ℓ, then (B_{w,ρ}, ON_{w}f)-bimodule X_{w} induces a Morita equivalence between ON_{w}f and B_{w,ρ}.

This result was proved by the second author independently [16].

Note that using this result, combining with [14],[17] and [3], one can prove Broué’s conjecture [1], [2] for weight two unipotent blocks of finite general linear groups.

In this paper, we consider the correspondences of various modules under the equivalence above. In section 3, we consider simple modules. In section 4, we treat Young modules and Specht type modules. Throughout the paper, we assume w < ℓ. The main results of this paper, namely, Corollary 3.3 and Theorem 4.9, enable us to calculate not only decomposition numbers of B_{w,ρ} but also the radical series of Specht type modules lying in B_{w,ρ}. These graded decomposition numbers are calculated in [15] explicitely in terms of the Littlewood-Richardson coefficients. (Moreover, we can also calculate the Loewy layers for Young modules lying in B_{w,ρ}.) The second author conjectured that under the condition “w < ℓ” these graded decomposition numbers lying in B_{w,ρ} are coincident with crystallized decomposition numbers introduced by [13] (see, also [11]). This conjecture is proved by [12].

Date: 2001.7.18.

2. Modules for the wreath product

In this paper, modules always mean finitely generated right modules, unless stated otherwise. Let λ ⊢ n. In [9], James defined a K[GL_{n}(q)]-module (resp. 𝕜GL_{n}(q)-module) S_{K}(λ) (resp. S_{𝕜}(λ)). S_{K}(λ) is simple and affords an irreducible unipotent character. On the other hand, S_{𝕜}(λ) has a unique simple quotient D(λ). Moreover, {D(λ) | λ ⊢ m, e-core of λ is ρ} is a complete set of representatives of isomorphic classes of simple 𝕜 ⊗ B_{w,ρ}-modules [5]. Let X_{O}(λ) be the Young OGL_{n}(q)-module corresponding to λ [6]. Then X_{O}(λ) is an indecomposable direct summand of a permutation OGL_{n}(q)-module induced from a parabolic subgroup. For R ∈ {K, 𝕜} let X_{R}(λ) be R ⊗_{O} X_{O}(λ). Let λ = (λ_{0}, …, λ_{e−1}) be a multipartition of w. Let R ∈ {K, O, 𝕜}. For each λ_{i}, we write S_{R}^{λ_{i}} for the Specht R𝔖_{|λ_{i}|}-module corresponding to λ_{i}. Note that, since w < ℓ, S_{𝕜}^{λ_{i}} is simple projective. Hence there is a unique projective O𝔖_{|λ_{i}|}-module S_{O}^{λ_{i}} which lifts S_{𝕜}^{λ_{i}}. Moreover, S_{K}^{λ_{i}} = K ⊗S_{O}^{λ_{i}}. Let ν_{i} = (i+1, 1^{e−1−i}) ⊢ e for 0 ≤ i ≤ e−1. Let T_{R} be one of X_{K}, X_{𝕜}, S_{K}, S_{𝕜} and D. Then T_{R}(ν_{i})^{⊗|λ_{i}|}⊗S_{R}^{λ_{i}} is an R[GL_{e}(q)≀𝔖_{|λ_{i}|}]- module [10]. We set,

T̃_{R}(λ) = Ind_{GL_{e}(q)≀𝔖λ}^{GL_{e}(q)≀𝔖_{w}}(⨂_{i=0}^{e−1}(T_{R}(ν_{i})^{⊗|_{λ_{i}}|} ⊗ S_{R}^{λ_{i}}) ⊗ S_{R}(ρ)

where S_{O}(ρ) is a projective indecomposable B_{0,ρ}-module (note that B_{0,ρ} has defect zero), S_{R}(ρ) = R ⊗ S_{O}(ρ) is a simple R[N_{w}]f-module for R = K, 𝕜. Then

{D̃(λ) | λ : e-tuple partition of w}

is a complete set of isomorphism classes of simple 𝕜N_{w}f-modules.

Let P(ρ, w) be the set of all partitions of m with e-core ρ. Let λ ∈ P(ρ, w). Using the abacus representation of ρ mentioned in Introduction, we can consider the e-quotient of λ [10]. We denote the e-quotient of λ by

λ̄ = (λ^{(0)}, …, λ^{(e−1)}).

The correspondence

λ ↔ λ̄

gives a bijection between P(ρ, w) and the set of all e-tuples partitions of w. Hence,

{D̃(λ̄) | λ ∈ P(ρ, w)}

is a complete set of representatives of isomorphism classes of simple 𝕜N_{w}f-modules. For α ∈ P (ρ, w − 1), let

Γ(α, i) = {λ ∈ P(ρ, w) | α^{(i)} ⊆ λ^{(i)}, λ^{(j)} = α^{(j)} (i ≠ j)}.

On the other hand, let P(w) be the set of all partitions of w and let P_{α}(w) = {λ ∈ P(w) | α ⊂ λ} for α ⊢ w − 1. We need the following combinatorial lemma.

Lemma 2.1. (1) Let σ and τ be bijections from P(w) to P(w). Suppose w > 2. If σ(P_{α}(w)) = τ(P_{α}(w)) for any α ⊢ w − 1, then σ = τ.

(2) Let σ and τ be bijections from P(ρ, w) to P(ρ, w). Suppose w > 2. If σ(Γ(α, i)) = τ(Γ(α, i)) for any α, i, then σ = τ.

Proof. (1) Let λ = (λ_{1}, λ_{2}, ⋯) ⊢ w. Suppose that λ_{i} > λ_{i+1} ≥ ⋯ ≥ λ_{j} > 0 for some i ≠ j. We define two partitions α, β as follows:

α = (α_{1}, α_{2}, ⋯), α_{i} = λ_{i} − 1, α_{k} = λ_{k} (k ≠ i)
β = (β_{1}, β_{2}, ⋯), β_{j} = λ_{j} − 1, β_{k} = λ_{k} (k ≠ j).

Then P_{α}(w) ∩ P_{β}(w) = {λ}, and σ(λ) = τ(λ).

So we may assume that λ = (λ_{1}^{m}). Let γ = (λ_{1}^{m−1}, λ_{1} − 1). Then σ(μ) = τ(μ) for μ ∈ P_{γ}(w) other than λ by the above argument. Hence we have σ(λ) = τ(λ).

(2) Let λ ∈ P(ρ, w). If there exists i such that λ^{(j)} = ∅ for any j ≠ i, then σ(λ) = τ(λ) by (1). So we may assume that λ^{(i)} ≠ ∅, λ^{(j)} ≠ ∅ for some i ≠ j. Take α, β ∈ P(ρ, w − 1) satisfyng:

α^{(i)} ⊂ λ^{(i)}, β^{(j)} ⊂ λ^{(j)}.

Then Γ(α, i) ∩ Γ(β, j) = {λ}, and σ(λ) = τ(λ). □

3. Simple modules

Definition Groups: Let G be GL_{n}(q) and let G′ be a Levi subgroup of G corresponding to a

composition (e, n − e), which is isomorphic to GL_{e}(q) × GL_{n−e}(q). Let N = N_{w}(resp. N′) be the normalizer of a Levi subgroup corresponding to a composition (e^{w}, |ρ|) in G(resp. G′), which is isomorphic to GL_{e}(q) ≀ 𝔖_{w} × GL_{|ρ|}(q) (resp. GL_{e}(q) × N_{w−1}).

Blocks: Let B_{0} be the principal block of OGL_{e}(q). Let b be the block idempotent of B_{w,ρ}. Let f (resp. f′) be the the block idempotent of B_{0}(O[GL_{e}(q) ≀ 𝔖_{w}]) ⊗ B_{0,ρ} (resp. B_{0} ⊗ B_{0}(O[GL_{e}(q) ≀ 𝔖_{w−1}]) ⊗ B_{0,ρ} ). For a block ideal of OH for some subgroup H of G we denote 𝕜 ⊗ B by B̄.

Functors: For any Levi subgroup L of G, we denote by R_{L}^{G} (resp. ^{∗}R_{L}^{G}) the Harish-Chandra induction (resp. restriction ) functor for G and L. Let X_{w}^{∨} be Hom_{B_{w,ρ}}(X_{w}, B_{w,ρ}). Define functors F_{w}, (F_{w})_{K} and F̄_{w} as follows:

F_{w} = − ⊗_{ONf} X_{w}^{∨} : mod-ONf ⟶ mod-B_{w,ρ}

(F_{w})_{K} = − ⊗_{K[N]f} (K ⊗ X_{w}^{∨}) : mod-K[N]f ⟶ mod-K ⊗ B_{w,ρ}

F̄_{w} = − ⊗_{𝕜Nf̄} X̄_{w}^{∨} : mod-𝕜Nf̄ ⟶ mod-B̄_{w,ρ}

For F ∈ {F_{w}, (F_{w})_{K}, F̄_{w}}, we denote the inverse of F by F^{∗}.

Lemma 3.1. Res_{G′×N′}^{G×N}(X_{w}) ≅ b R_{G′×N′}^{G×N}(B_{0} ⊗ X_{w−1}).

Proof. First,

Ind_{N′×N′}^{G×N′}(ON′f′) ≅ R_{G′×N′}^{G×N′}(B_{0} ⊗ X_{w−1}) ⊕ V

where V is a direct sum of indecomposable modules with vertex not conjugate of ΔD. On the other hand, Res_{G×N′}^{G×N}(X_{w}) is a direct summand of Ind_{N′×N′}^{G×N′}(ON′f′). Since Res_{G×N′}^{G×N}(X_{w}) is indecomposable by [19], Res_{G×N′}^{G×N}(X_{w}) is a direct summand of R_{G′×N′}^{G×N′}(B_{0} ⊗ X_{w−1}). Since

(F_{w})_{K}(Ind_{N′}^{N}(K[N′]f′))
≅ b(R_{G′}^{G})(id ⊗ (F_{w−1})_{K})(K[N′]f′)

by [19],

K ⊗ X_{w} ≅ b(R_{G′×N′}^{G×N′}(K ⊗ (B_{0} ⊗ X_{w−1}))

as K[G]-modules. Hence

Res_{G′×N′}^{G×N}(X_{w}) ≅ b R_{G′×N′}^{G×N}(B_{0} ⊗ X_{w−1}).
□

In order to compare the labels of some B̄_{w,ρ}-modules with that of some 𝕜Nfmodules, the following definition will be important.

Definition Let λ̌ ∈ P(ρ, w) be the partition such that

λ̌^{(i)} = \{ \begin{matrix}λ^{(i)} & if e + i : odd, \\ λ^{(i)}′ & if e + i : even. \end{matrix}

Theorem 3.2. (F_{w})_{K}(S̃_{K}(λ̄)) ≅ S_{K}(λ̌).

Proof. We proceed by induction on w. The case w = 2 is proved in [19]. So we assume w > 2. Define a bijection σ : P(ρ, w) ⟶ P(ρ, w) by

(F_{w})_{K}(S̃_{K}(λ̄)) ≅ S_{K}(σ(λ)).

It suffices to show that σ(λ) = λ̌. By the Littlewood-Richardson rule,

(F_{w})_{K}(Ind_{N′}^{N}S_{K}(ν_{i}) ⊗ S̃_{K}(ᾱ)) = (F_{w})_{K}( ⨁_{μ∈Γ(α,i)} S̃_{K}(μ̄)) = ⨁_{μ∈Γ(α,i)} S_{K}(σ(μ))

for α ∈ P(ρ, w−1) and 0 ≤ i ≤ e − 1. On the other hand,

b R_{G′}^{G}(id ⊗ (F_{w−1})_{K})(S_{K}(ν_{i}) ⊗ S̃_{K}(ᾱ)) = b R_{G′}^{G}(S_{K}(ν_{i}) ⊗ S_{K}(α̌))

= ⨁_{μ∈Γ(α̌,i)} S_{K}(μ) = ⨁_{μ∈Γ(α,i)} S_{K}(μ̌)

by induction and [19]. Since these two modules are isomorphic by Lemma 3.1, we have σ(Γ(α, i)) = (Γ(α, i))^{∨}. Hence σ(λ) = λ̌ by Lemma 2.1. □

Corollary 3.3. F̄_{w}(D̃(λ̄)) ≅ D(λ̌) for λ ∈ P(ρ, w).

Proof. Let λ ∈ P(ρ, w). We will show that F̄_{w}(D̃(¯λ̌)) ≅ D(λ) by induction on λ. Suppose that F̄_{w}(D̃(¯μ̌)) ≅ D(μ) for any μ ∈ P(ρ, w), μ > λ. Suppose that F̄_{w}(D̃(¯ν̌)) ≅ D(λ) for ν ∈ P(ρ, w). We write V ↔ W if modules V and W have the same composition factors. Then

S_{𝕜}(λ) ↔ (⨁_{μ>λ} d_{λμ}D(μ)) ⊕ D(λ)

for some nonnegative integers d_{λμ}. By Theorem 3.2,

S̃_{𝕜}(¯λ̌) ↔ (⨁_{μ>λ} d_{λμ}D̃(¯μ̌)) ⊕ D̃(¯ν̌).

Since S̃_{𝕜}(¯λ̌) has D̃(¯λ̌) as a composition factor by definition, we have ν = λ. □

4. Young modules

Recall that

X̃_{R}(λ) = Ind_{GL_{e}(q)≀𝔖λ}^{GL_{e}(q)≀𝔖_{w}}(⨂_{i=0}^{e−1} X_{R}(ν_{i})^{⊗|λ_{i}|} ⊗ S_{R}^{λ_{i}}) ⊗ S_{R}(ρ)

and S̃_{R}(λ) is a submodule of X̃_{R}(λ) by definition.

Lemma 4.1. [X̃_{𝕜}(λ) : D̃(λ)] = 1.

Proof. By Mackey’s decomposition theorem, we know

Res_{GL_{e}(q)^{×w}}^{GL_{e}(q)≀𝔖_{w}} Ind_{GL_{e}(q)≀𝔖λ}^{GL_{e}(q)≀𝔖_{w}}(V) ≅ ⨁_{g∈[𝔖λ\𝔖_{w}]}(Res_{GL_{e}(q)^{×w}}^{GL_{e}(q)≀𝔖λ}(V))^{g}

for any R[GL_{e}(q) ≀ 𝔖_{λ}]-module V. Here, [𝔖_{λ}\𝔖_{w}] is a set of the right coset representatives of 𝔖_{λ}\𝔖_{w}.

In particular,

Res_{GL_{e}(q)^{×w}}^{GL_{e}(q)≀𝔖_{w}}(D(λ))

(1)  ≅ (∏_{i=0}^{e−1} dim_{R} S_{𝕜}^{λ_{i}}) · ⨁_{g∈[𝔖λ\𝔖_{w}]} (⨂_{i=0}^{e−1}(D(ν_{i}))^{⊗|λ_{i}|})^{g} ⊗ S_{𝕜}(ρ).

Comparing the composition factors of Res_{GL_{e}(q)^{×w}}^{GL_{e}(q)≀𝔖λ} ⨂_{i=0}^{e−1} X_{𝕜}(ν_{i})^{⊗|λ_{i}|}⊗S_{𝕜}^{λ_{i}} with

⨂_{i=0}^{e−1} D(ν_{i})^{⊗|λ_{i}|}, we get [Res_{GL_{e}(q)^{×w}}^{GL_{e}(q)≀𝔖λ} ⨂_{i=0}^{e−1} X_{𝕜}(ν_{i})^{⊗|λ_{i}|}⊗S_{𝕜}^{λ_{i}} : ⨂_{i=0}^{e−1} D(ν_{i})^{⊗|λ_{i}|}] =

∏_{i=0}^{e−1} dim_{𝕜} S_{𝕜}^{λ_{i}}. Therefore, by (1) we have [X̃_{𝕜}(λ) : D̃(λ)] = 1. □

Let η_{λ} (resp. χ_{λ}) be the character afforded by X̃_{O}(λ) (resp. S̃_{K}(λ)).

For e-tuple multipartitions λ and μ we define a total order λ ≻ μ by

λ ≻ μ if there exists m such that λ_{m} > μ_{m} and λ_{j} = μ_{j} for j > m.

Then, by definition clearly the following holds:

Proposition 4.2. ⟨η_{λ}, χ_{λ}⟩ = 1 and ⟨η_{λ}, χ_{μ}⟩ = 0 if λ ≻ μ.

In particular, we know the following:

Corollary 4.3. If λ ≠ μ, then X̃_{𝕜}(λ) ≅ ˜X̸_{𝕜}(μ).

Lemma 4.4. Soc(X̃_{𝕜}(λ)) = Soc(S̃_{𝕜}(λ))

Proof. Since the functor Ind_{GL_{e}(q)≀𝔖λ}^{GL_{e}(q)≀𝔖_{w}} preserves the Loewy layers of modules,

Soc(X̃_{𝕜}(λ)) = Ind_{GL_{e}(q)≀𝔖λ}^{GL_{e}(q)≀𝔖_{w}}(⨂_{i=0}^{e−1}(Soc(X_{𝕜}(ν_{i})))^{⊗|λ_{i}|} ⊗ S_{𝕜}^{λ_{i}}) ⊗ S_{𝕜}(ρ).

Similarly, we have

Soc(S̃_{𝕜}(λ)) = Ind_{GL_{e}(q)≀𝔖λ}^{GL_{e}(q)≀𝔖_{w}}(⨂_{i=0}^{e−1}(Soc(S_{𝕜}(ν_{i})))^{⊗|λ_{i}|} ⊗ S_{𝕜}^{λ_{i}}) ⊗ S_{𝕜}(ρ).

However, Soc(X_{𝕜}(ν_{i})) = Soc(S_{𝕜}(ν_{i})) for i = 0, …, e − 1. Hence, we are done. □

Since S̃_{𝕜}(λ) is indecomposable, by Lemma 4.1 we immediately know

Corollary 4.5. X̃_{𝕜}(λ) is indecomposable.

Lemma 4.6. F̄_{w}^{∗}(X_{𝕜}(λ)) ≅ X̃_{𝕜}(ν̄) for some ν ∈ P(ρ, w).

Proof. Let L := L_{(e^{w},ρ)} be the Levi subgroup of G corresponding to the composition (e^{w}, ρ) ⊧ m. Res_{L}^{N}(F̄_{w}^{∗}(V)) is a direct summand of b_{w} ·^{∗}R_{L}^{G}(V) for any B̄_{w,ρ}-module V. Any indecomposable direct summand of Res_{L}^{N}(F̄_{w}^{∗}(X_{𝕜}(λ))) has the following shape :

⨂_{i=1}^{w} X(ν_{m(i)}) ⊗ S_{𝕜}(ρ).

Here, m(i) is an element of {0, 1, 2, …, e − 1} for any i ∈ {1, 2, …, w}. Hence, any indecomposable direct summand of F̄_{w}^{∗}(X_{𝕜}(λ)) is a direct summand of

(Ind_{L}^{N}(⨂_{i=0}^{e−1} X(ν_{i})^{⊗n_{i}})) ⊗ S_{𝕜}(ρ)

for some (n_{0}, n_{1}, …, n_{e−1}) ⊧ w. Since F̄_{w}^{∗} is an equivalence, we get

F̄_{w}^{∗}(X_{𝕜}(λ)) ≅ X̃_{𝕜}(ν) for some ν.
□

Theorem 4.7. F̄_{w}(X̃_{𝕜}(λ̄)) ≅ X_{𝕜}(λ̌) for any λ ∈ P(ρ, w).

Proof. It suffices to show that

(2)  F̄_{w}^{∗}(X_{𝕜}(λ)) ≅ X̃_{𝕜}(¯λ̌)

for any λ ∈ P(ρ, w). We proceed by induction on (>, P(ρ, w)). By Corollary 3.3, we have already shown (2) for the maximal element of P(ρ, w). Suppose that the claim (2) holds for any μ ⪈ λ. By Lemma 4.6 there exists ν ∈ P(ρ, w) such that

F̄_{w}^{∗}(X_{𝕜}(λ)) ≅ X̃_{𝕜}(¯ν̌).

In particular, X̃_{𝕜}(¯ν̌) must have D̃(¯ν̌) as a composition factor. So, we deduce by Corollary 3.3 that X_{𝕜}(λ) ≅ F̄_{w}(X̃_{𝕜}(¯ν̌)) must have D(ν) ≅ F̄_{w}(D̃(¯ν̌)) as a composition factor. Hence, by [8, 16.3] we deduce ν ⊵ λ. Suppose that ν ⪈ λ. By the assumption of induction, we have

F̄_{w}^{∗}(X_{𝕜}(ν)) ≅ X̃_{𝕜}(¯ν̌).

In other word, we have

F̄_{w}^{∗}(X_{𝕜}(ν)) ≅ F̄_{w}^{∗}(X_{𝕜}(λ)).

Therefore, we have ν = λ and get a contradiction. □

Let P (λ) (resp. P̃(λ̄)) be the projective indecomposable module corresponding to D(λ) (resp. D̃(λ̄)). Then, Specht type modules S_{𝕜}(λ), S̃_{𝕜}(λ̄) and Young modules X_{𝕜}(λ), X̃_{𝕜}(λ̄) enjoy the following properties:

Lemma 4.8. (1) If τ ∈ Hom_{𝕜G}(P (λ), X_{𝕜}(λ)) is such that

Top(τ(P (λ))) ≅ D(λ), then τ(P (λ)) ≅ S_{𝕜}(λ).

(2) If ψ ∈ Hom_{𝕜N}(P̃(λ̄), X̃_{𝕜}(λ̄)) is such that Top(ψ(P̃(λ̄))) ≅ D̃(λ̄), then ψ(P̃(λ̄)) ≅ S̃_{𝕜}(λ̄).

Proof. (1) is clear by [6]. (2): By Lemma 4.1 we have already known [X̃_{𝕜}(λ̄) : D̃(λ̄)] = 1. Moreover, S̃_{𝕜}(λ̄) is a submodule of X̃_{𝕜}(λ̄) and have its unique top D̃(λ̄). So, ψ(P̃(λ̄)) must be isomorphic to S̃_{𝕜}(λ̄). □

By Corollary 3.3, Theorem 4.7 and Lemma 4.8, we have

Theorem 4.9. F̄_{w}(S̃_{𝕜}(λ̄)) ≅ S_{𝕜}(λ̌) for any λ ∈ P(ρ, w).

Ackowledgments

We must thank T. Okuyama for noticing us his method for Broué’s abelian defect conjecture and bringing to us J. Chuang’s work of blocks of symmetric groups with weight 2. We must thank R. Dipper and G. Hiss for letting us notice J. Chuang and R. Kessar’s work. In particular, G. Hiss told the second author the details of J. Chuang and R. Kessar’s work when we worked at the unipotent blocks with weight 2. We also thank J. Chaung and R. Kessar for sending their preprint to our supervisor S. Koshitani.

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Department of Mathematics, Faculty of Education, Saitama University, Urawa-ku, Saitama 338, Japan

e-mail: ahida@post.saitama-u.ac.jp

Department of Mathematics Graduate School of Science and Technology Science University of Tokyo Yamazaki, Noda, Chiba, 278-8510 Japan

e-mail: miyachi@ma.noda.sut.ac.jp