On maximal actions and w-maximal actions of finite hypergroups

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DOI 10.1007/s10801-007-0082-3

On maximal actions and w-maximal actions of finite hypergroups

Bangteng Xu

Received: 11 September 2006 / Accepted: 30 May 2007 / Published online: 13 June 2007

Abstract Sunder and Wildberger (J. Algebr. Comb. 18, 135–151,2003) introduced the notion of actions of finite hypergroups, and studied maximal irreducible actions and *-actions. One of the main results of Sunder and Wildberger states that if a finite hypergroupKadmits an irreducible action which is both a maximal action and a *- action, thenK arises from an association scheme. In this paper we will first show that an irreducible maximal action must be a *-action, and hence improve Sunder and Wildberger’s result (Theorem 2.9). Another important type of actions is the so-called w-maximal actions. For aw-maximal actionπ:K→Aff(X), we will prove thatπ is faithful and|X| ≥ |K|, and|K|is the best possible lower bound of|X|. We will also discuss the strong connectivity of the digraphs induced by aw-maximal action.

Keywords Hypergroups·Association schemes·Actions·Maximal actions,

*-actions·w-maximal actions

1 Introduction

In this paper we study the maximal actions andw-maximal actions of finite hypergroups. Due to the strong similarities between the algebraic structures of finite hypergroups and Bose-Mesner algebras of association schemes, we hope that the study of finite hypergroups will bring a different point of view for the study of association schemes. Actions of finite hypergroups on a finite set, introduced by Sunder and Wild- berger [8], provide a way to establish direct connections between finite hypergroups and association schemes. Sunder and Wildberger [8] proved that a finite hypergroup Karises from an association scheme ifKadmits an irreducible action which is both a maximal action and a *-action. It is well-known that the study of C-algebras and

B. Xu (

⁾

Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475, USA

e-mail: [email protected]

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table algebras, whose algebraic structures are very similar to the algebraic structure of finite hypergroups, has many interesting applications to association schemes; for example, see [1,4,7], and [10].

Hypergroups have been studied by many researchers in various fields for a long time; for references, see [6] or [9]. An actionπ of a finite hypergroupK on a finite setXassigns to everyc_i∈Kan affine mapπ(c_i), which is identified with a column stochastic matrix whose rows and columns are indexed by elements inX, such that the productπ(c_i)π(c_j)is a linear combination of{π(c_i)|c_i ∈K}with coefficients equal to the structure constants ofc_ic_j. Sunder and Wildberger [8] introduced the notion of irreducible actions, maximal irreducible actions, and *-actions. One of their main results in [8] states that if a finite hypergroupK admits an irreducible action which is both a maximal action and a *-action, then K arises from an association scheme. In this paper we will first show that an irreducible maximal action must be a

*-action, and hence improve Sunder and Wildberger’s result [8, Theorem 2.9]. Then we introduce the concept ofw-maximal actions, a broad class of actions that includes maximal actions and left regular actions. For aw-maximal actionπ :K→Aff(X), we will prove that π is faithful and|X| ≥ |K|, and|K|is the best possible lower bound of|X|. We will also discuss the strong connectivity of the digraphs induced by aw-maximal action, and obtain another point of view to a well-known fact in the theory of association schemes.

The rest of this introductory section gives notation, definitions, theorems, and examples. Throughout this paper,Cdenotes the complex numbers,R⁺the positive real numbers, andNthe positive integers.

Definition 1.1 A finite hypergroup is a distinguished linear basisK= {c0, c1, . . . , c_n} of a complex unital associative *-algebraCKthat satisfies the following conditions, for 0≤i, j≤n:

(i) c0is the multiplicative identity ofCK;

(ii) c_ic_j=_n

k=0n^k_ijc_k, wheren^k_ij∈R⁺∪ {0}for allk, and_n

k=0n^k_ij=1;

(iii) K is a self-adjoint set, i.e., there exists an involutive mapping i →i^∗ of {0,1, . . . , n}such thatc_i∗=c^∗_i; and

(iv) n⁰_ij>0 if and only ifj=i^∗, wheren⁰_ij is the same as in (ii).

Let K = {c0, c1, . . . , cn} be a finite hypergroup, with cicj =_n

k=0n^k_ijck, 0≤ i, j ≤n. Then (n⁰_i_∗_i)⁻¹ is called the weight of ci, and denoted by w(ci). Note that for any i, w(ci)=w(c^∗_i) [8, Lemma 1.3]. The weight of K is defined by w(K):=_n

i=0w(c_i).

There are many important examples of finite hypergroups, including the following one.

Example 1.2 LetXbe a finite set, and(X,{Ri}0≤i≤n)be an association scheme over X(not necessarily commutative). LetA_i be the adjacency matrix with respect toR_i, andn_i be the valency ofR_i. ThenK:= {n⁻_i ¹A_i |0≤i≤n}is a finite hypergroup (see [4, Section 2.2]). Note that the weight of n⁻_i ¹Ai is ni, and the weight ofK is|X|.

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The above example is the motivation of the next definition.

Definition 1.3 Let K = {c0, c1, . . . , cn} be a finite hypergroup with cicj= _n

k=0n^k_ijc_k, 0≤i, j≤n. If there is an association scheme (X,{R_i}0≤i≤n)over a finite setXwith adjacency matricesA_i and valenciesn_i, 0≤i≤n, such that

(n⁻_i ¹A_i)· (n⁻_j¹A_j)= n k=0

n^k_ij(n⁻_k¹A_k), 0≤i, j≤n,

then we say that the finite hypergroup K arises from the association scheme (X,{R_i}0≤i≤n).

If a finite hypergroup K = {c0, c1, . . . , cn} arises from an association scheme (X,{R_i}0≤i≤n), using the same notation as in Definition1.3, we have thatw(c_i)=n_i, for alli, andw(K)= |X|.

LetX= {x1, x2, . . . , x_k}be a finite set, andV be the vector space over Rwith basisX. The simplex ofXis the subset

sX:= {a1x1+a2x2+ · · · +a_kx_k∈V |a_i≥0, for alli,anda1+a2+ · · · +a_k=1}. An affine map (or a convex map) ofsXis a mapT :sX→sXthat satisfies

T (λα+(1−λ)β)=λT (α)+(1−λ)T (β), for allα, β∈sX, λ∈ [0,1]. The set of all affine maps ofsXis denoted by Aff(X). LetT ∈Aff(X). Then for any α1, α2, . . . , α_t ∈sXand anyλ1, λ2, . . . , λ_t ∈ [0,1]withλ1+λ2+ · · · +λ_t=1, we have that

T (λ1α1+λ2α2+ · · · +λ_tα_t)=λ1T (α1)+λ2T (α2)+ · · · +λ_tT (α_t).

For any T ∈Aff(X), letT (x_i)=_k

j=1λ_{j i}x_j ∈sX, for all i, and let Mat(T ) be thek×kmatrix whose rows and columns are indexed byx1, x2, . . . , x_k and whose (xi, xj)-entry is λij. Then Mat(T ) is a column-stochastic matrix (i.e., the sum of every column is 1), and T is uniquely determined by Mat(T ). Furthermore,T → Mat(T )is a bijection between Aff(X)and the set of column-stochastick×kmatrices.

For the rest of the paper, we will always identify Aff(X) with the set of column- stochastick×kmatrices whose rows and columns are indexed by the elements ofX.

Furthermore, for any matrixA∈Aff(X)and anyx, y∈X, the (x, y)-entry of A is denoted byAx,y.

Definition 1.4 ([8, Definition 2.1]) LetK= {c₀, c₁, . . . , c_n}be a finite hypergroup withc_ic_j=_n

k=0n^k_ijc_k, 0≤i, j≤n. An action ofKon a finite setXis a mapping π:K−→Aff(X), c_i→π(c_i)

such that

(i) π(c0)=I, the identity matrix, and (ii) π(c_i)π(c_j)=_n

k=0n^k_ijπ(c_k), 0≤i, j≤n.

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Sunder and Wildberger [8] introduced the concepts of *-actions, irreducible actions, and maximal irreducible actions. Let π :K→Aff(X)be an action of a finite hypergroupKon a finite setX. Thenπ is called a *-action if for anyc_i∈K, π(c_i^∗)=π(c_i)^∗, the Hermitian adjoint matrix ofπ(c_i). (But here since every entry of π(c_i)is a real number,π(c_i)^∗is the transpose matrix ofπ(c_i).) If for anyx, y∈X, there isc_i ∈K such that the (x, y)-entry ofπ(c_i),π(c_i)_x,y, is not zero, then π is called an irreducible action. For an irreducible actionπ : K→Aff(X), by [8, The- orem 2.6(i)] we see that

|X| ≤w(K).

Definition 1.5 ([8, Definition 2.8]) An irreducible actionπ:K→Aff(X)of a finite hypergroupKon a finite setXis called a maximal action if|X| =w(K).

Example 1.6 If a finite hypergroupK= {c₀, c₁, . . . , c_n}arises from an association scheme(X,{R_i}0≤i≤n), using the same notation as in Definition1.3, then it is clear that

π:K→Aff(X), ci→n⁻_i ¹Ai

is an irreducible action which is both a *-action and a maximal action.

One of the main results in [8] states that if a finite hypergroupK admits an irreducible action which is both a maximal action and a *-action, thenKarises from an association scheme. Our first main result shows that an irreducible maximal action must be a *-action.

Theorem 1.7 LetKbe a finite hypergroup andπ:K→Aff(X)be an irreducible action. Ifπis a maximal action, thenπis a *-action.

From Theorem1.7, Theorem 2.9 in [8] can be improved as follows.

Theorem 1.8 A finite hypergroupKarises from an association scheme if and only if Kadmits an irreducible maximal action.

Theorems1.7and1.8will be proved in Section2.

Now let us introduce the concept of aw-maximal action. LetK= {c₀, c₁, . . . , c_n} be a finite hypergroup. Then

e₀:=w(K)⁻¹ n i=0

w(c_i)c_i∈CK (1.1)

is the Haar measure ofK, and ([8, (1.5)])

e₀=e^∗₀=e²₀=c_ie₀=e₀c_i, for alli. (1.2) Letπ:K→Aff(X)be an action. Thenπcan be linearly extended to a map from the convex hull ofKinCK,co(K), to Aff(X). (Note thatco(K)can be identified in

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a natural manner with the simplexsK.) Thus, π(e0)=w(K)⁻¹

n i=0

w(c_i)π(c_i), (1.3)

andπ(e₀)is an idempotent matrix. Furthermore, by [8, Proposition 2.3],π is irreducible if and only if for any x, y∈X,π(e₀)_x,y=π_x, where everyπ_x∈R⁺, and

x∈Xπ_x=1.

Let π : K → Aff(X) be an irreducible action, with π(e0)x,y =πx, for all x, y∈X, wheree0is the Haar measure ofK. Then the weight ofx∈Xwith respect to the actionπis defined by

w_π(x)= π_x miny∈Xπ_y.

Sowπ(x)≥1 for allx∈X. The weight of the setX with respect to the actionπ is defined by

wπ(X)=

x∈X

wπ(x)= 1 miny∈Xπ_y. By [8, Theorem 2.6(i)], we have that

|X| ≤wπ(X)≤w(K). (1.4)

This is the justification of the next definition.

Definition 1.9 An irreducible actionπ:K→Aff(X)of a finite hypergroupKon a finite setXis called aw-maximal action ifw_π(X)=w(K).

Clearly any maximal action is aw-maximal action. But aw-maximal action need not to be a maximal action (see Example3.1in Section3). Example3.1also reveals that every finite hypergroup has aw-maximal action. Note that ifπ:K→Aff(X)is a *-action for a finite hypergroupK, thenπis maximal if and only ifπisw-maximal (Lemma3.2in Section3).

Letπ:K→Aff(X)be an action, and let MatX(C)be the set of all square matrices overCwhose rows and columns are indexed by the elements ofX. Thenπ can be linearly extended to an algebra homomorphism (still denoted by π) from CK to MatX(C). The algebra homomorphismπ:CK→MatX(C)is a *-homomorphism if π is a *-action.

Definition 1.10 An actionπ :K→Aff(X)is called faithful if the algebra homo- morphismπ:CK→MatX(C)is injective.

So an actionπ:K→Aff(X)is faithful if and only if{π(ci)|ci∈K}is a linearly independent subset in MatX(C).Our next main result is the following

Theorem 1.11 LetKbe a finite hypergroup andπ:K→Aff(X)be aw-maximal action. Thenπis faithful and|X| ≥ |K|.

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Example3.1in Section3indicates that|K|is the best possible lower bound of|X| for aw-maximal actionπ:K→Aff(X). Also note that a faithful irreducible action need not to be aw-maximal action (see Example 3.6in Section3).

We will prove Theorem 1.11 in Section3. We will also introduce the digraphs induced by an action of a finite hypergroup, which are similar to the digraphs induced by an association scheme, and discuss the strong connectivity of these digraphs in Section3. In particular, the study of strong connectivity of the digraphs induced by a w-maximal action yields another point of view to a well-known fact in the theory of association schemes.

2 Maximal actions

In this section we will prove Theorems1.7and 1.8. Let us show two lemmas first.

Lemma 2.1 LetK= {c0, c1, . . . , c_n}be a finite hypergroup, and letπ:K→Aff(X) be an irreducible action. Then the following are equivalent.

(i) π is a maximal action.

(ii) π(e₀)=w(K)⁻¹J_|_X_|, wheree₀is the Haar measure ofK, andJ_|_X_|is the|X| ×

|X|matrix whose entries are all 1.

(iii) For anyx∈X,π(c_i)_x,x=0, for alli =0.

Proof (i) ⇒(ii) Since π is irreducible, by [8, Proposition 2.3], for all x, y∈X, π(e0)x,y=πx, whereπx>0, and

x∈Xπx=1. Note that for anyx∈X,wπ(x)= π_x/min_y∈Xπ_y≥1, andw(K)= |X| ≤

x∈Xw_π(x)=w_π(X)≤w(K)by [8, The- orem 2.6(i)]. So we get thatw_π(x)=1 for allx∈X. Hence allπ_x,x∈X, are equal.

Thusπx= |X|⁻¹, for allx∈X, andπ(e0)= |X|⁻¹J_|X|. But|X| =w(K), So (ii) holds.

(ii)⇒(iii) For anyx∈X,w(K)⁻¹=π(e₀)_x,xand (1.3) imply that w(K)⁻¹=w(K)⁻¹

n i=0

w(c_i)π(c_i)_x,x

=w(K)⁻¹w(c0)π(c0)x,x+w(K)⁻¹ n i=1

w(ci)π(ci)x,x

=w(K)⁻¹+w(K)⁻¹ n i=1

w(c_i)π(c_i)_x,x.

Since everyw(c_i) >0 and everyπ(c_i)_x,x≥0, we have that π(c_i)_x,x=0, for allx∈X,1≤i≤n.

So (iii) holds.

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(iii)⇒(i) Since for anyx∈Xand anyi =0,π(c_i)_x,x=0, we see that π(e0)x,x=w(K)⁻¹

n i=0

w(ci)π(ci)x,x=w(K)⁻¹, for allx∈X.

Hence, [8, Proposition 2.3] implies that π(e₀)_x,y =π(e₀)_x,x =w(K)⁻¹, for all x, y∈X. Thus,π(e0)=w(K)⁻¹J_|_X_|. Butπ(e0)is an idempotent matrix of size|X|, we must have thatw(K)= |X|. Thus,πis a maximal action, and (i) holds.

Letπ:K→Aff(X)be a maximal action. Then Lemma2.1implies that

πx=π(e0)x,y= |X|⁻¹=w(K)⁻¹, for allx, y∈X. (2.1) Lemma 2.2 LetK= {c₀, c₁, . . . , c_n}be a finite hypergroup, and letπ:K→Aff(X) be an action. Then the following hold.

(i) For anyx∈X,

[π(ci)π(c^∗_i)]x,x≥w(ci)⁻¹, 0≤i≤n. (2.2) (ii) Ifπis an irreducible action, withπ(e₀)_x,y=π_x, for allx, y∈X, wheree₀is the

Haar measure ofK, then

0≤π(c_i)_x,y≤w(K)π_x

w(c_i) , for allx, y∈X,0≤i≤n. (2.3) Proof (i) Assume thatcicj=_n

k=0n^k_ijck, 0≤i, j≤n. Then for anyci,π(ci)π(c^∗_i)

=_n

k=0n^k_ii_∗π(c_k).Hence, [π(c_i)π(c^∗_i)]x,x=

n k=0

n^k_ii_∗π(c_k)_x,x≥n⁰_ii_∗π(c₀)_x,x=w(c_i^∗)⁻¹.

Sincew(ci)⁻¹=w(c_i^∗)⁻¹by [8, Lemma 1.3], (2.2) holds.

(ii) For anyiand anyx, y∈X,π_x=π(e₀)_x,yyields that πx=w(K)⁻¹

n j=0

w(cj)π(cj)x,y≥w(K)⁻¹w(ci)π(ci)x,y, for alli.

Hence, 0≤π(c_i)_x,y≤w(K)π_xw(c_i)⁻¹, and (2.3) holds.

Now we are ready to prove Theorem1.7.

Proof of Theorem1.7 Assume that K= {c₀, c₁, . . . , c_n}, with c_ic_j =_n

k=0n^k_ijc_k, 0≤i, j≤n. Letπ(e0)_x,y=π_x, for allx, y∈X. Note that for anyx∈X,w(K)π_x= 1 by (2.1). So by Lemma2.2(ii),

0≤π(c_i)_x,y≤w(c_i)⁻¹, for allx, y∈X,0≤i≤n. (2.4)

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Now we claim that

for anyx, y∈X, π(c^∗_i)_y,x =0⇒π(c_i)_x,y =0, 0≤i≤n. (2.5) If (2.5) is not true, then there are ci and x, y ∈X such that π(c^∗_i)y,x =0 but π(c_i)_x,y=0. Recall that the sum of every column ofπ(c_i^∗)is 1. So

z∈Xπ(c^∗_i)_z,x

=1. Hence (2.4) forces that [π(ci)π(c_i^∗)]x,x=

z∈X

π(ci)x,zπ(c_i^∗)z,x=

z∈X,z =y

π(ci)x,zπ(c^∗_i)z,x

≤w(c_i)⁻¹

z∈X,z =y

π(c^∗_i)_z,x

< w(c_i)⁻¹

z∈X

π(c_i^∗)_z,x

=w(c_i)⁻¹.

That is,[π(c_i)π(c^∗_i)]x,x< w(c_i)⁻¹, a contradiction to (2.2). Thus, (2.5) must hold.

Note that for anyi,(c^∗_i)^∗=c_i. So (2.5) yields that

for anyx, y∈X, π(ci)x,y =0 ⇐⇒ π(c^∗_i)y,x =0, for alli. (2.6) Next we show that

for anyx, y∈X, π(c_i)_x,y =0⇒π(c_j)_x,y=0, for allj =i. (2.7) If (2.7) is not true, then there are x, y ∈X and c_i, c_j such that π(c_i)_x,y =0, π(cj)x,y =0, andj =i. So by (2.6),π(c_j^∗)y,x =0. Hence,

[π(c_i)π(c_j^∗)]x,x=

z∈X

π(c_i)_x,zπ(c_j^∗)_z,x≥π(c_i)_x,yπ(c^∗_j)_y,x>0.

But on the other hand,π(c_i)π(c_j^∗)=_n

k=0n^k_ij_∗π(c_k),n⁰_ij_∗=0, and Lemma2.1force that

[π(ci)π(c^∗_j)]x,x= n k=0

n^k_ij∗π(ck)x,x= n k=1

n^k_ij∗π(ck)x,x=0, a contradiction. This proves (2.7).

Letx, y∈X. Ifπ(c_i)_x,y =0, then for anyj =i,π(c_j)_x,y=0 by (2.7). Recall thatw(K)⁻¹=π(e0)_x,y by Lemma2.1. Hence,

w(K)⁻¹=w(K)⁻¹ n j=0

w(c_j)π(c_j)_x,y=w(K)⁻¹w(c_i)π(c_i)_x,y.

Soπ(c_i)_x,y=w(c_i)⁻¹. Thus, for anyx, y∈Xand anyc_i, we have eitherπ(c_i)_x,y= 0 orπ(c_i)_x,y=w(c_i)⁻¹. Sincew(c^∗_i)⁻¹=w(c_i)⁻¹, (2.6) yields thatπ(c^∗_i)=π(c_i)^∗, for alli. Soπis a *-action, and the theorem is proved.

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The proof of Theorem1.8is very easy now. We include a proof here for the con- venience of readers. A similar proof can be found in [8]. From Example1.2, we only need to prove that if a finite hypergroupK admits an irreducible maximal action π:K→Aff(X), thenKarises from an association scheme. Using the same notation as in the proof of Theorem1.7, letA_i:=w(c_i)π(c_i), 0≤i≤n. Then everyA_i is a (0,1)-matrix, and

(i) A0=I, the identity matrix,

(ii) A₀+A₁+ · · · +A_n=J, the matrix whose entries are all 1, (iii) A^∗_i =A_i∗, 0≤i≤n, and

(iv) A_iA_j= n k=0

w(ci)w(cj)n^k_ij

w(ck) A_k,0≤i, j≤n.

(Note that since allA_i are (0,1)-matrices, andA₀, A₁, . . . , A_n are linearly independent, everyw(c_i)w(c_j)n^k_ij/w(c_k)is a nonnegative integer, 0≤i, j, k≤n.) For anyi∈ {0,1, . . . , n}, define a relationRionXby

(x, y)∈Ri ⇐⇒ (Ai)x,y =0, for allx, y∈X.

Then(X,{Ri}0≤i≤n)is an association scheme. For anyi,Ai is the adjacency matrix with respect to R_i, and w(c_i) is the valency ofR_i. Therefore, K arises from the association scheme(X,{R_i}0≤i≤n). This proves Theorem1.8.

3 w-maximal actions

In this section we will present a characterization ofw-maximal actions, and use this characterization to prove Theorem1.11. Then we study the strong connectivity of the digraphs induced by aw-maximal action. The so-calledπ-minimal point plays an important role in our discussion.

Let π :K→Aff(X) be an irreducible action. Then by (1.4), |X| ≤w_π(X)≤ w(K). Clearly ifπ is a maximal action, thenπ is also aw-maximal action. But the next example shows that aw-maximal action need not to be a maximal section. This example also shows that every finite hypergroup has aw-maximal action.

Example 3.1 LetK= {c₀, c₁, . . . , c_n}be a finite hypergroup, withc_ic_j=_n

k=0n^k_ijc_k, 0 ≤i, j ≤n. Let X =K. Then the left-regular action of K is defined by π : K →Aff(X), π(c_i)_c_j_,c_k =n^j_ik. π is indeed an action of K by the associativity of multiplication in CK. Let e0 be the Haar measure of K. By (1.1) and (1.2), e0ci=e0=w(K)⁻¹_n

j=0w(cj)cj, for alli. Hence,

π(e₀)=w(K)⁻¹

⎛

⎜⎜

⎝

w(c₀) w(c₀) · · · w(c₀) w(c₁) w(c₁) · · · w(c₁)

... ... . .. ... w(cn) w(cn) · · · w(cn)

⎞

⎟⎟

⎠.

Since everyw(c_i)≥1 andw(c₀)=1,π is an irreducible action by [8, Proposition 2.3], andw_π(X)=w(K). Soπ is aw-maximal action. However,π need not to be

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a maximal action.π is maximal if and only if everyw(c_i)=1 if and only ifKis a group.

Theorem1.11states that for aw-maximal actionπ:K→Aff(X),|X| ≥ |K|. The above example reveals that|K|is the best possible lower bound for|X|.

For irreducible *-actions, maximal andw-maximal are identical. This is the next lemma.

Lemma 3.2 Let K be a finite hypergroup and π:K→Aff(X)be an irreducible

*-action. Thenπ is maximal if and only ifπ isw-maximal.

Proof Since π is an irreducible *-action,π(e₀)= _|_X¹_|J, where J is the|X| × |X| matrix whose entries are all 1. Thus w_π(x)=1, for all x ∈X. So w_π(X)=

x∈Xw_π(x)= |X|. Henceπ is maximal if and only if it isw-maximal.

The next lemma gives a very useful characterization ofw-maximal actions. This characterization will be needed for the rest of our discussion.

Lemma 3.3 LetK= {c₀, c₁, . . . , c_n}be a finite hypergroup andπ:K→Aff(X)be an irreducible action. Thenπis aw-maximal action if and only if there existsx∈X such thatπ(c_i)_x,x=0, for alli =0.

Proof Recall thatπ(e₀)=w(K)⁻¹_n

i=0w(c_i)π(c_i), and for anyx, y∈X,π(e₀)_x,y

=π_x, where everyπ_x∈R⁺, and

x∈Xπ_x=1. If π is aw-maximal action, then w(K)=w_π(X)=_min_y¹_∈_X_π_y. So there isx∈Xsuch thatπ_x=w(K)⁻¹. But

π_x=π(e0)_x,x=w(K)⁻¹ n i=0

w(c_i)π(c_i)_x,x

=w(K)⁻¹+w(K)⁻¹ n i=1

w(ci)π(ci)x,x.

Hence,π(ci)x,x=0, for alli =0.

On the other hand, if there existsx∈Xsuch thatπ(ci)x,x=0, for alli =0. Then as above,πx=π(e0)x,x=w(K)⁻¹. Hence

wπ(X)= 1

miny∈Xπ_y ≥ 1

π_x =w(K).

But by [8, Theorem 2.6(i)], w_π(X)≤w(K). Thus,w_π(X)=w(K), andπ is w-

maximal.

LetK= {c₀, c₁, . . . , c_n}be a finite hypergroup,π :K→Aff(X)aw-maximal action, andx∈X such thatπ(c_i)_x,x=0, for alli =0. Thenπ_x=w(K)⁻¹≤π_y, for ally∈X. Because of the importance of such anx, we introduce the following definition.

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Definition 3.4 LetK= {c₀, c₁, . . . , c_n}be a finite hypergroup andπ:K→Aff(X) be aw-maximal action. Then anyx∈Xsuch thatπ(c_i)_x,x=0, for alli =0, is called aπ-minimal point.

Now we are ready to prove Theorem1.11.

Proof of Theorem1.11 Assume thatK= {c0, c1, . . . , cn}, withcicj=_n

k=0n^k_ijck, 0≤i, j≤n. Letx0∈Xbe aπ-minimal point. Then

π(c_i)_x₀_,x₀=0,for alli =0 and w(K)=w_π(X)=(π(e0)_x₀_,y)⁻¹,for ally∈X.

(3.1) Let us first show that|X| ≥ |K|. Note that (3.1) yields that

[π(c^∗_j)π(c_i)]x0,x0 = n k=0

n^k_j_∗_iπ(c_k)_x₀_,x₀=n⁰_j_∗_i=δ_ijw(c_i)⁻¹, 0≤i, j≤n, (3.2) whereδij is the Kronecker delta. By (3.2) we can prove the following claims.

Claim 1: For anyi, there exists at least oney∈Xsuch thatπ(ci)y,x₀ =0.

If Claim 1 is not true, then there is aci such that for anyy∈X,π(ci)y,x₀ =0.

Hence [π(c_i^∗)π(ci)]x₀,x₀ =

y∈Xπ(c^∗_i)x₀,yπ(ci)y,x₀ =0, a contradiction to (3.2).

So Claim 1 must hold.

Claim 2: For anyiandy∈X, ifπ(ci)y,x₀ =0, thenπ(c^∗_i)x₀,y =0.

Note thatw(K)πx₀=1 by (3.1). Hence, Lemma2.2(ii) yields that

0≤π(ci)x0,y≤w(ci)⁻¹, for ally∈X,0≤i≤n. (3.3) If Claim 2 is not true, then there are ci and y ∈X such that π(ci)y,x₀ =0 but π(c_i^∗)x₀,y=0. Recall that

z∈Xπ(ci)z,x₀=1. Hence (3.3) forces that [π(c^∗_i)π(c_i)]x0,x0 =

z∈X

π(c^∗_i)_x₀_,zπ(c_i)_z,x₀=

z∈X,z =y

π(c_i^∗)_x₀_,zπ(c_i)_z,x₀

≤w(c^∗_i)⁻¹

z∈X,z =y

π(ci)z,x₀

< w(c^∗_i)⁻¹

z∈X

π(ci)z,x₀

=w(c^∗_i)⁻¹.

That is, [π(c^∗_i)π(ci)]x₀,x₀ < w(c_i^∗)⁻¹=w(ci)⁻¹, a contradiction to (3.2). Thus, Claim 2 must hold.

Claim 3: For anyiandy∈X, ifπ(ci)y,x₀ =0, then for allj =i, π(cj)y,x₀=0.

If Claim 3 does not hold, then there arei, j, i =j, andy∈Xsuch thatπ(ci)y,x₀ = 0 andπ(cj)y,x₀ =0. Hence by Claim 2,π(c^∗_j)x₀,y =0. Therefore,

[π(c^∗_j)π(c_i)]x0,x0=

z∈X

π(c^∗_j)_x₀_,zπ(c_i)_z,x₀≥π(c^∗_j)_x₀_,yπ(c_i)_y,x₀>0, a contradiction to (3.2). So Claim 3 holds.

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Recall that every entry ofπ(e₀)=w(K)⁻¹_n

i=0w(c_i)π(c_i)is positive. So for all y∈X, there isc_i such thatπ(c_i)_y,x₀ =0. Hence, we can define a map

X−→K, y→ci ifπ(ci)y,x₀ =0.

This map is well-defined by Claim 3, and is surjective by Claim 1. Therefore,|X| ≥

|K|.

The faithfulness ofπ also follows from Claims 1 and 3. Here is another direct short proof. Ifπ is not faithful, then someπ(c_r)is a linear combination of{π(c_i)| i =r}. Assume thatπ(c_r)=

i =rα_iπ(c_i). Thenπ(c_r)π(c^∗_r)=

i =rα_iπ(c_i)π(c^∗_r).

Therefore,

n k=0

n^k_rr_∗π(c_k)=

i =r

α_i n k=0

n^k_ir_∗π(c_k).

Note thatn⁰_ir_∗=0 for alli =r. So (3.1) implies that n⁰_rr∗=

n k=0

n^k_rr∗π(ck)x₀,x₀=

i =r

αi

n k=0

n^k_ir∗π(ck)x₀,x₀ =

i =r

αin⁰_ir∗=0,

a contradiction. Hence,πmust be faithful.

From the proof of Theorem1.11, we have the following

Corollary 3.5 Let K= {c0, c1, . . . , c_n}be a finite hypergroup,π:K→Aff(X)a w-maximal action, andx0∈Xaπ-minimal point. Then the following hold.

(i) For anyy∈X, there is a uniquec_i such thatπ(c_i)_y,x₀ =0.

(ii) For anyy∈X, ifπ(c_i)_y,x₀ =0, thenπ(c^∗_i)_x₀_,y =0.

The next example reveals that a faithful action need not to be aw-maximal action.

Example 3.6 LetK= {c₀, c₁, c₂}be a finite hypergroup with c²₁=1

3c0+2

3c2, c1c2=c2c1=c1, c²₂=1 2c0+1

2c2, andc_i^∗=ci,for alli.

Then the actionπ:K→Aff(X), whereX= {x₁, x₂, x₃, x₄}, defined by

π(c₀)=

⎛

⎜⎝

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎞

⎟⎠, π(c₁)=1 2

⎛

⎜⎝

0 0 1 1

1 1 0 0

⎞

⎟⎠,

π(c2)=¹₄

⎛

⎜⎝

1 3 0 0

3 1 0 0

0 0 1 3

0 0 3 1

⎞

⎟⎠

is a faithful irreducible action. Note that everyπ(c_i)is symmetric. Soπis a *-action.

Hencewπ(X)= |X| =4<6=w(K). Therefore,π is not aw-maximal action.

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Let K= {c₀, c₁, . . . , c_n}be a finite hypergroup. For anya=_n

i=0α_ic_i ∈CK, define Supp(a)= {c_i |α_i =0}. A nonempty subsetS of K is a subhypergroup of K if for anyc_i, c_j ∈S, Supp(cic_j)⊆S. IfSis a subhypergroup ofK, thenc₀∈S, S is closed under the involution * ofK, andS itself is a hypergroup. Note that for anyci∈S, the weight ofci as an element in S is the same as the weight ofci as an element inK. Sow(S)≤w(K), andw(S)=w(K)if and only ifS=K.

Let S be a subhypergroup of K, and π:K →Aff(X) be an action. Then, via restriction,π induces an action

π_S:S→Aff(X), c_i→π(c_i),for allc_i∈S, called the restriction ofπtoS.

Proposition 3.7 LetSbe a proper subhypergroup ofK, andπ:K→Aff(X)be a w-maximal action. Then the restrictionπ_Sis not an irreducible action.

Proof Toward a contradiction, assume that π_S is an irreducible action. Let K = {c₀, c₁, . . . , c_n}, andx₀∈Xbe aπ-minimal point. Then for anyi =0,π(c_i)_x₀_,x₀=0.

Hence, for any ci ∈S with i =0, πS(ci)x₀,x₀ =π(ci)x₀,x₀ =0. So πS is also a w-maximal action. For anyci ∈K, there exists y∈X such that π(ci)y,x₀ =0 by Claim 1 in the proof of Theorem1.11. SinceπS is irreducible, there iscj∈S such thatπ_S(c_j)_y,x₀ =0. Soπ(c_j)_y,x₀ =0, and we must have thatc_i=c_j∈Sby Corol- lary3.5(i). Hence, everyc_i ∈K is also inS, andK=S, a contradiction to the as- sumption thatS is a proper subhypergroup ofK. Therefore,π_Sis not an irreducible

action.

Proposition3.7is not necessarily true if the actionπ is notw-maximal, as indi- cated by the next example.

Example 3.8 LetKbe the same as in Example3.6. Then the actionπ:K→Aff(X), whereX= {x1, x2}, defined by

π(c0)= 1 0

0 1

, π(c1)=1 2

1 1 1 1

, π(c2)=1 4

1 3 3 1

is an irreducible action. Note thatwπ(X)=2<6=w(K). Soπis not aw-maximal action. LetS= {c0, c2}. ThenSis a proper subhypergroup ofK, and the restriction action

π_S:S→Aff(X), c₀→π(c₀), c₂→π(c₂)

is also irreducible. It is also interesting to note that althoughSis a proper subhypergroup ofK,

w_π_S(X)=2=w_π(X).

Similar to the digraphs induced by association schemes (see [3] or [4]), the digraphs induced by actions of hypergroups can be defined as follows. Let K =