Wreath products and projective system of non Schurian association schemes

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, with an explicit description of primitive idempotents.


Introduction
Association schemes are central objects in algebraic combinatorics, with many interactions with other areas of mathematics.The wreath product is a method to construct an association scheme from two association schemes.This notion appears in the monograph [16, P.45-47] by Weisfeiler, in a more general context of coherent configurations (in the terminology of cellular algebras).Song [15] gives a description in terms of association schemes and their adjacency matrices.Muzychuk [13] defines the wreath product of association schemes and its generalizations (he used the term "wedge product").There is a construction of non-Schurian S-rings by using a generalized wreath product for S-rings, studied by Evdokimov and Ponomarenko [5].
In this paper, we construct projective systems of non-Schurian association schemes in Section 4. As preliminaries, we recall some basic facts on wreath products in Section 2 with a self-contained proof.In Section 3, we shall study the automorphism group of wreath products and their Schurian property.A large part of the results in Section 3.2 is covered by more detailed discussions in the monograph by Chen and Ponomarenko [3, Sections 3.2 and 3.4], but we give proofs for self-containedness, e.g., on the known fact that the wreath product is Schurian if and only if both components are Schurian ( [3,Corollary 3.4.7]).This yields a large class of non-Schurian association schemes and non-Schurian S-rings, from the known examples.
In Section 4, we consider iterated wreath products.The kernel scheme by Martin-Stinson [11] is an example.An infinite iterated wreath product gives a projective system of association schemes, namely, profinite association schemes in the sense of [12], with an explicit description of their primitive idempotents.In [12], only Schurian examples are given.One of our motivations is to give examples of projective systems of non-Schurian association schemes.

Wreath products
2.1.Category of association schemes.Let us recall the notion of association schemes briefly.See Bannai-Ito [1] and Delsarte [4] for details.We summarize basic terminologies.
Definition 2.1.Let X be a finite set.By #X we denote the cardinality of X.Let C(X) denote the vector space of mappings from X to C. By the multiplication of functions, C(X) is a unital commutative ring.The set C(X × X) is naturally identified with the set of complex square matrices of size #(X), and the matrix product is given by AB(x, z) = y∈X A(x, y)B(y, z).The Hadamard product • is given by the component-wise product, namely, (A • B)(x, z) = A(x, z)B(x, z).(Note that • may denote the composition of mappings, but no confusion would occur.)Definition 2.2.Let X, I be finite sets, and R : X × X → I a surjection.We call (X, R, I) an association scheme, if the following properties (1), (2), and (3) are satisfied.For each i ∈ I, R −1 (i) may be regarded as a relation on X, denoted by R i .Let A i be the corresponding adjacency matrix in C(X × X).The surjection R induces an injection C(I) → C(X × X).Let A X ⊂ C(X × X) be the image.
(1) There is an i 0 ∈ I such that A i0 is an identity matrix.
(2) A X ⊂ C(X × X) is closed under the matrix product.
(3) A X is closed under the transpose of C(X × X).The algebra A X with the two multiplications (i.e. the Hadamard product and the matrix product) is called the Bose-Mesner algebra of (X, R, I).The set of A i (i ∈ I) is the set of primitive idempotents with respect to Hadamard products, and each is called an Hadamard primitive idempotent (or an adjacency matrix).We call X the underlying set of the association scheme, and I the set of the relations.The number #I − 1 is called the number of classes.We may use the same notation i 0 for distinct association schemes.The number of 1 in a row of A i is independent of the choice of the row, called the i-th valency and denoted by k i .It is also the number of 1 in each column of A i .If A X is commutative with respect to the matrix product, then (X, R, I) is said to be commutative.
In the following, we write simply an "association scheme X" for an association scheme (X, R, I) by an abuse of language.Definition 2.3.For a commutative association scheme X, it is known that A X with matrix product is isomorphic to the direct product of #I copies of C as a ring.Elements corresponding to (0, . . ., 0, 1, 0, . . ., 0) are called the primitive idempotents.We denote by J = J(X) the set of primitive idempotents.Let E j denote the primitive idempotent corresponding to j ∈ J.The notation j 0 is kept for E j0 = 1 #X J, where J denotes the matrix whose components are all 1.We use the symbol j 0 for any commutative association schemes.
We follow MacLane [10] for the terminologies of the category theory, in particular, isomorphisms, functors, projective systems, and limits.The association schemes form a category, by the following (e.g.Hanaki [6] and Zieschang [18]).Definition 2.4.Let (X, R, I) and (X ′ , R ′ , I ′ ) be association schemes.A morphism of association schemes from X to X ′ is a pair of functions f : X → X ′ and σ : I → I ′ such that the following diagram commutes: By the commutativity, σ preserves i 0 .It is clear that the surjectivity of f implies that of σ.The morphism (f, σ) is said to be surjective if f is surjective.
2.2.Wreath products.Although originally the wreath product is introduced by Weisfeiler [16, P.45], this section follows Song [15, §4], since his description is due to association schemes and convenient for our purpose.The definition, a proof, and the eigenmatrix of wreath products are all given there.We recall these, partly because we use some different symbols and notations which make the definition and proof simpler, and partly because we use some part of the proof in the following arguments.
Definition 2.5.Let (X, R X , I X ) and (Y, R Y , I Y ) be association schemes.Then, is an association scheme, called the direct product of X and Y [15, §3] (called the tensor product in [16]).In fact, this is the direct product in the category of association schemes.
It is easy to see the following.
Proposition 2.6.The adjacency matrices of the direct product are If both X and Y are commutative, then so is X × Y , and the primitive idempotents are The wreath product is defined as follows.The non-standard symbol ⋋ comes from the "L" of lexicographic.Definition 2.7.Define lex : We denote Definition 2.8.For a mapping σ : I X → I X preserving i 0 ∈ I X and a mapping τ : by σ on I X \ {i 0 } and by τ on I Y .(See Definition 2.4 for the preservation of i 0 .)We also define Remark 2.9.This remark is for category-oriented readers, and may be skipped since it is not essential in this paper.Let Sets !be the category of sets with a base point, namely, an object is a set with a base point and a morphism is a mapping preserving the base points.In other words, for a fixed singleton {!}, Sets ! is the category of morphisms {!} → S with S being a set (a special case of comma categories).We remark that I X ⋋I Y gives the coproduct functor Sets where × : Sets !× Sets !→ Sets ! is the direct product functor.On the other hand, there is a canonical natural transformation given by , and i y ∈ I Y → (i 0 , i y ).

Consequently, if pr
is the projection functor to the i-th component (i = 1, 2), then there is a natural transformation ⋋ ⇒ pr i for each i = 1, 2, and π IX is the evaluation of ⋋ ⇒ pr 1 at (I X , I Y ).
Definition 2.10.(Wreath products) Let (X, R X , I X ) and (Y, R Y , I Y ) be association schemes.Then the composition is an association scheme, which is called the wreath product of X and Y , and denoted by X⋋Y.
Remark 2.11.The underlying set of X⋋Y is X × Y , whereas the set of relations is I X ⋋I Y with the cardinality #I X + #I Y − 1.The wreath product is a fusion of the direct product.
We first thought that the name "wreath product" seems misleading, since this seems to have no relation with the wreath product of groups (as Song [15] remarked in his introduction).However, there is a strong relation [16, P.47, §4.8] via transitive permutation groups.Still, we point out the similarity to the "lexicographic ordering," since the above definition means that the relation of two elements in X × Y is determined by first looking at the X-components, and decided according to their relation when they are different, and otherwise by the relation at the Ycomponents.This procedure is similar to the lexicographic ordering.The next is a translation of [15,Theorem 4.1].
(1) For association schemes X and Y , their wreath product X⋋Y is an association scheme.(2) Let A iX (i X ∈ I X ) (and A iY (i Y ∈ I Y ), respectively) be the adjacency matrices of X (those of Y , respectively).Then, the adjacency matrices of X⋋Y are as follows.
Type front: where I denotes the identity matrix (of size #X) and J denotes the matrix whose components are all 1 (square of size #Y ).
Proof.Take an element i To check that these constitute an association scheme, we only need to show that this set of matrices is closed under the transpose and contains the unit, which are obvious, and that matrix products of these matrices are linear combinations of these.A product of the front-type matrices is a linear combination of those, since X is an association scheme.A product of the rear-type matrices is a linear combination of those, since Y is an association scheme.For the mixed case, by JA iY = k iY J, the product is a scalar multiple of a matrix of the front-type.
Here we start our own observations.Corollary 2.13.The wreath product X⋋Y is not P Proof.The wreath product is P -polynomial, if there is an adjacency matrix that generates the Bose-Mesner algebra as a ring over C. By #X > 1, we have an i X = i 0 .By #Y > 1, we have an i Y = i 0 .The linear span of the front-type matrices is identical with the set A X ⊗ J and is closed under the matrix multiplication.The linear span of the rear-type matrices is identical with the set I ⊗ A Y and is closed under the matrix multiplication.Thus, a front-type adjacency matrix does not generate a rear-type adjacency matrix I ⊗ A iY , since i Y = i 0 and hence A iY = J.A rear-type adjacency matrix does not generate a front-type adjacency matrix Corollary 2.14.The wreath product of association schemes is a commutative association scheme if and only if both components are commutative.
Proof.This follows from the above description of the Hadamard primitive idempotents.
Corollary 2.15.Suppose that both X and Y are commutative.Let E jX (j X ∈ J X ) (and E jY (j Y ∈ J Y ), respectively) be the primitive idempotents of X (of Y , respectively).Then, the primitive idempotents of X⋋Y are as follows.
Type front: Thus, Proof.One sees that these elements are linear combinations of the adjacency matrices in Theorem 2.12 because X and Y are commutative association schemes, and hence in the Bose-Mesner algebra of X⋋Y .It is easy to check that these are idempotents and the product of any two distinct elements is zero, and hence are linearly independent.Since the number of these matrices is #X + #Y − 1, which is the cardinality of I X ⋋I Y , these idempotents span the Bose-Mesner algebra.
Corollary 2.16.Let X and Y be commutative association schemes.Then, X⋋Y Proof.A similar proof to that of Corollary 2.13 applies to.
Corollary 2.17.We use where P iX (j X ) is the eigenvalue of A iX for eigenvector E jX , and P iY (j Y ) is the eigenvalue of A iY for eigenvector E jY .Proof.This follows by a diagram chasing.

Automorphisms and Schurian property
3.1.Automorphism group of a wreath product.Definition 3.1.For an association scheme X, we denote by Aut(X) the group of automorphisms in the category of association schemes (Definition 2.4).On the other hand, for any set X equipped with any structure (such as an association scheme or a group), Aut Sets (X) denote the group of bijections from X to X, neglecting the structure of X.
As stated before, the wreath product of association schemes is studied and generalized by Muzychuk [13], where in Proposition 3.2 the automorphism group of generalized wreath products of thin-association schemes is determined.Chen and Ponomarenko [3, Theorem 3.4.6]determined the automorphism group of the wreath product, but their definition of automorphism groups is the normal subgroup Aut(X|I) < Aut(X) in Lemma 3.5 below, and thus different from ours. 1e give a description of the automorphism group of wreath products of association schemes.
Theorem 3.2.Let X and Y be association schemes, and p : X × Y → X be the projection.Then, for any (f, σ) ∈ Aut(X⋋Y ), f maps each fiber Y x := p −1 (x) to another fiber.Thus, f induces an element π(f ) ∈ Aut Sets (X).This gives a surjective morphism of groups with a natural splitting morphism.Thus, we have a group isomorphism where K := ker P , whose structure is described in Proposition 3.3 below.We have a natural embedding of Aut(Y ) into K that acts trivially on X and diagonally on each fiber Y x .The image of the embedding commutes with the image of the splitting.Thus, Aut(X⋋Y ) has a subgroup isomorphic to Aut(Y ) × Aut(X).
Proof.We construct P .An automorphism in Aut(X⋋Y ) is a pair of bijections commute.We claim that τ permutes I X \ {i 0 } and I Y separately.Take an element i X ∈ I X \ {i 0 }.Then the corresponding valency is k iX #Y ≥ #Y , by Theorem 2.12.For i Y ∈ I Y , the valency is k iY ≤ #Y , with equality holds only if I Y = {i 0 }.Thus, there is no automorphism that maps i X to i Y .Thus we have π(τ ) which makes the following commute: For (x, y) and (x, y ′ ), since This means that the X-component of f (x, y) is the same with that of f (x, y ′ ), i.e., f maps a fiber of p to another fiber, which shows the unique existence of π(f ) that makes a commutative diagram Now we have where the commutativity of the two triangles follows from Proposition 2.18.This commutativity, (3.3), and the surjectivity of the projection and hence gives an automorphism of the association scheme X, which gives the group homomorphism P .Conversely, an automorphism (g, τ ) of X, namely, a pair of bijections g : X → X, τ : I X → I X with an additional commutativity condition gives a pair of bijections g × id Y : X × Y → X × Y and τ ⋋id IY : I X ⋋I Y → I X ⋋I Y (see Definition 2.8), which is an automorphism of the association scheme X⋋Y , splitting the homomorphism P .On the other hand, it is easy to show that any automorphism (g, σ) of Y induces an automorphism (f, τ ) of X⋋Y , by putting f = id X × g and τ = id IX ⋋σ (see Definition 2.8), which lies in the kernel of P and commutes with the image of the splitting of P .
On the kernel of P , we show the following.Proposition 3.3.The kernel K of P : Aut(X⋋Y ) → Aut(X) is a subgroup of x∈X Aut(Y x ).More precisely, we have where T is the image in Aut Sets (I Y ) of Aut(Y ), and Aut(Y x ) τ denotes the set of elements in Aut(Y x ) whose Aut Sets (I Y )-component is τ .(Note that for (f, τ ) ∈ Aut(Y ), f uniquely determines τ , hence the above union is disjoint.)Thus, there is an injective group homomorphism where the right-hand side is the wreath product of groups with respect to the action of Aut(X) on X.
Proof.We compute the kernel K of P .It is clear that Take any (k, σ) ∈ K and x ∈ X, and look the action on Y x .Clearly σ = (id IX ⋋(σ| IY )).There exists k x ∈ Aut Sets (Y ) such that k(x, y) = (x, k x (y)).For any implies that (k x , σ| IY ) ∈ Aut(Y ).This holds for any x ∈ X, and thus Conversely, take an element from the right-hand side of (3.6): Thus, k x shares τ ∈ Aut Sets (I Y ) for any x such that (k x , τ ) ∈ Aut(Y ).We claim that (k, id IX ⋋τ ) ∈ K.In fact, for (x 1 , y 1 ) and (x 2 , y 2 ), if and if follows.The rest of the claims, i.e., the relation to the wreath product, holds at the level of permutation groups.That is, the set of permutations of The detail is omitted.

Construction of non-Schurian schemes.
The results of this subsection are known to the researchers in this area, and proofs are given in Chen and Ponomarenko [3, Sections 3.2 and 3.4] with more systematic and detailed studies.However, we felt that it might not easy to follow the details, and decided to include our proofs, for the reader's convenience.

Definition 3.4. (Schurian association schemes)
Let G be a group and X a set with left transitive action of G. Then the quotient by the diagonal action R : X × X → G\(X × X) =: I is known to be a (possibly non-commutative) association scheme.An association scheme isomorphic to this type is called a Schurian association scheme.
We may replace G with its image in Aut Sets (X).Then, a pair (g, id I ) for g ∈ G is an automorphism of the Schurian scheme, and G transitively acts on R −1 (i) for each i ∈ I.The following is immediate from this observation.Lemma 3.5.Let (X, R, I) be an association scheme.Let Aut(X|I) ⊂ Aut(X) be the subgroup consisting of elements that act on I trivially.Then, X is Schurian if and only if Aut(X|I) acts transitively on R −1 (i) for each i ∈ I.

The next is [3, Corollary 3.2.22].
Proposition 3.6.Let X and Y be association schemes.The following are equivalent.
(1) Both X and Y are Schurian.
(2) The direct product X × Y is Schurian.
Proof.Assume (1).Then, it is easy to check that we have a natural group homomorphism Aut(X|I For any (i X , i Y ) ∈ I X × I Y , the subset acts transitively by the assumption, and by Lemma 3.5, we conclude (2).
Conversely, assume (2).By symmetry, it suffices to show that X is Schurian.We claim that there is a natural group morphism holds.Thus p(f (x, y)) = p(f (x, y ′ )), hence we have π(f ) that makes the diagram (3.3) commute.Take any y ∈ Y .If we denote I X × I Y → I X by the same symbol p, we have Thus, Aut(X|I X ) transitively acts on R −1 X (i X ), which shows (1).The next is [3,Corollary 3.4.7].
Theorem 3.7.Let X and Y be association schemes.The following are equivalent.
(1) Both X and Y are Schurian.
Assume (1).Take four points in X × Y such that Assume that . By the last statement of Theorem 3.2, we have 3 lies in K, and since Together with (3.8), we conclude (1).
Corollary 3.8.The wreath product of a non-Schurian association scheme and an association scheme (in both order of product) is non-Schurian.The same statement holds for the direct product.
Thus, there exists a large family of non-Schurian schemes.We remark that there are substantial studies on construction of non-Schurian schemes, e.g., Evdokimov-Ponomarenko [5], Hanaki-Hirai-Ponomarenko [7], and Hirasaka-Kim [8].Non-Schurian Schur rings are of particular interest since historically Wielandt [17,Theorem 26.4] found such an example, answering a question by Schur.To avoid confusion, we use the term S-rings for Schur rings (see Definition 3.10 below).
The results for S-rings stated in the rest of this section are closely related with the results by Evdokimov-Ponomarenko [5].They used generalized wreath products to construct non-Schurian S-rings in a cyclic group, using delicate arguments.We deal with only direct products of groups and the usual wreath products, but still give a construction of non-Schurian S-rings.
We start with a definition of Cayley association schemes, which is equivalent to the notion of S-rings.We denote by e the unit of a group.We use the terminologies such as S-rings and Schurian S-rings according to a survey by Muzychuk and Ponomarenko [14].The following definition of Cayley association schemes is given in [1, II.6] (without naming), as well as the equivalence to the notion of S-rings.We changed g 2 g −1 1 in the definition there to g −1 1 g 2 because we consider the left action.A detailed study on Cayley association schemes and S-rings is found in [3, Section 2.4].

Definition 3.9. (Cayley association schemes)
Let G be a finite group.If there is a surjective mapping r : 2 ) is an association scheme, then it is called a Cayley association scheme.This notion is equivalent to the following notion of S-rings.The conditions on r : G → I are equivalent, and the Bose-Mesner algebra of a Cayley association scheme is naturally isomorphic to the corresponding S-ring.

Let
A r = the C-linear span of r −1 (i) ⊂ C[G] for i ∈ I .
Then A r is called an S-ring, if the following conditions are satisfied.
(2) A r is closed under the product in C[G].
(3) For any i ∈ I, there is We want to discuss on the Schurian property of S-rings.
Definition 3.11.(Schurian S-rings) Let X be a finite set.Let Γ be a group transitively acting on X, with a subgroup G acting transitively and faithfully on X. Fix x ∈ X.Then we have a bijection G → X, g → gx, and through this bijection Γ acts on G. Let Γ e be the stabilizer of e ∈ G. Consider r : G → Γ e \G =: I.
Then r : G → I gives an S-ring, which is called a Schurian S-ring.
The next proposition is well-known, and the proof is omitted.Proposition 3.12.Let G be a finite group, and assume that r : G → I gives a Cayley association scheme, and equivalently, an S-ring.Then, the following are equivalent.
(1) The Cayley association scheme is Schurian in the sense of Definition 3.4.
(2) The S-ring is Schurian in the sense of Definition 3.11.
The following proposition is a direct consequence of the definitions and the equivalence between Cayley association schemes and S-rings.Proposition 3.13.The wreath product of two S-rings is an S-ring.The direct product of two S-rings is an S-ring.
Proof.Let G 1 and G 2 be finite groups.Let r 1 : G 1 → I 1 and r 2 : G 2 → I 2 be the corresponding mappings.Then, their wreath product is given by where the top arrow is ((g 1 , g 2 ), (g 3 , g 4 )) → (g 1 , g 2 ) −1 (g 3 , g 4 ) = (g −1 1 g 3 , g −1 2 g 4 ).The definition of the wreath product is via the left bottom corner.By the commutativity of the diagram, it is an S-ring.The claim for the direct product follows in a similar manner, by merely removing I 1 ⋋I 2 from the above diagram.Proposition 3.12, Proposition 3.13 and Theorem 3.7 imply the following proposition.
Proposition 3.14.The wreath product of a non-Schurian S-ring and an S-ring (in both order) is non-Schurian.The same statement holds for the direct product.
The existence of a large number of non-Schurian S-rings follows.Using generalized wreath products, Evdokimov-Ponomarenko [5] proved the following theorem.As another example, Hanaki-Hirai-Ponomarenko [7] proved a generalization of Wielandt's construction: Theorem 3.16.Let p be a prime.Let G be an elementary abelian p-group of even rank except for the orders 2 2 , 3 2 , and 2 4 .Then G has a non-Schurian S-ring.
Starting from these examples, by taking the wreath product with any S-rings or association schemes, we have a large family of non-Schurian S-rings and non-Schurian association schemes.

Iterated product and profinite association schemes
A special case of an iterated wreath product is implicitly used in the construction of the kernel scheme by Martin-Stinson [11] (the notation here follows [12]).Definition 4.1.Let n be a positive integer, and V a finite set of alphabet with cardinality v ≥ 2. Let X n be V n , and I n := {1, 2, . . ., n} ∪{∞}.(We use ∞ instead of a natural notation n + 1, since this is i 0 and to be distinguished when considering a projective system in the next section.)Define R n : X n × X n → I n as follows.Let x = (x 1 , x 2 , . . ., x n ) and y = (y 1 , y 2 , . . ., y n ) be elements of X n .Let R(x, y) be the smallest index i for which x i = y i .If x = y, then R(x, y) = ∞.This is a symmetric (and hence commutative) association scheme, with R −1 (∞) being the identity relation.This is called a kernel scheme, and denoted by − −−− → k(n, v).
Definition 4.2.Let X be an association scheme.We define inductively its wreath power for n ∈ N by: • X ⋋1 := X.
Definition 4.3.Let v ≥ 2 be an integer.Let H(1, v) be the class-one association scheme of size v, namely, the unique association scheme with Proof.This can be proved in a straightforward manner by induction on n.The definition of X⋋Y is "to determine the relation of (x, y) and (x ′ , y ′ ), first look at the X-component; if x = x ′ then the relation is decided by them in X.If not, then the relation is decided by those of y and y ′ ."This is compatible with the definition of the kernel schemes.
Remark that in Martin-Stinson [11], the kernel schemes are shown to be association schemes by computing the intersection numbers.The above proposition gives another proof.which has Hadamard product with unit, and the convolution product (without unit if J ∧ is infinite).It has a linear basis J ∧ consisting of all the primitive idempotents, and is isomorphic, as a ring with Hadamard product, to the space of locally constant functions on I ∧ .For subsets J D ⊂ J ∧ , I C ⊂ I ∧ , and a finite multi-subset Y ⊂ X ∧ , a property "I C -free code and J D -design" of Y is defined.
Proof.Because of the definition of π IX : I X ⋋I Y → I X in Definition 2.8, the preimage of i X ∈ I X in I X × I Y is and hence the image of by Proposition 2.6.Since A X is the linear span of A iX , the statement follows.
Proposition 4.9.Let X 1 , X 2 , . . ., X n be any sequence of association schemes.Let I 1 , I 2 , . . ., I n be the set of their relations.Then, their wreath product and the set of relations Proof.The structure of the underlying set follows by definition.The structure of the set of relations follows by induction from Definition 2.7.
Then, the series of the wreath products form a projective system of association schemes.The mappings of the underlying sets are given by projections The mappings of the sets of relations are given by mapping the elements in I n to the i 0 in I n−1 .
The projective limit of the underlying set is the direct product (with direct product topology, hence compact and Hausdorff ) The projective limit I ∧ of I 1 ⋋ • • • ⋋I n is the one-point compactification of the discrete topological set ∞ i=1 (I i \ {i 0 }).
Proof.The mapping between the underlying set is the projection by Proposition 2.18.It is a general fact that the projective limit of finite direct products is the infinite direct product.
The projective system (I 1 ⋋ • • • ⋋I n ) n∈N>0 is given by mapping the last I n to the i 0 of I n−1 , by Proposition 2.18.We consider its projective limit.Except for i 0 , every element in the coproduct is a clopen point in the projective limit, and the set of open neighborhoods of the limit of i 0 is the set of the union of {i 0 } and the complement of a finite set of ∞ i=1 (I i \ {i 0 }).Proposition 4.11.Suppose that every X n is commutative in Proposition 4.10.Then, the primitive idempotents of Its inductive limit is

Proposition 2 . 18 .
Let X and Y be association schemes.The projection p : X × Y → X of the underlying sets and the mapping π IX : I X ⋋I Y → I X in Definition 2.8 give a surjective morphism of association schemes π : X⋋Y → X.
and let G x be the stabilizer of {x} × Y in G (i.e. the set of elements in G that preserve {x} × Y as a set).Hence G x acts on Y , and by definition of G, G x trivially acts on I Y and consequently G x ⊂ Aut(Y |I Y ) follows by the commutativity of

Definition 3 .
10. (S-rings) Let G be a finite group, and r : G → I a surjective mapping.Let C[G] be the group ring.For a subset S ⊂ G, define S := s∈S s ∈ C[G].

4. 2 .Lemma 4 . 7 .Lemma 4 . 8 .
Iterated wreath products.Iterated wreath products give examples of profinite association schemes.We begin with preparation.The wreath product is associative, i.e., there is a canonical isomorphism(X⋋Y )⋋Z → X⋋(Y ⋋Z)for association schemes X, Y and Z. Hence we may write X⋋Y ⋋Z.Proof.This follows from the identification(I X ⋋I Y )⋋I Z = ([(I X \ {i 0 }) I Y ] \ {i 0 }) I Z = (I X \ {i 0 }) (I Y \ {i 0 }) I Z = I X ⋋(I Y ⋋I Z )and the commutativity ofI X × I Y × I Z (I X ⋋I Y ) × I Z I X × (I Y ⋋I Z ) (I X ⋋I Y )⋋I Z = I X ⋋(I Y ⋋I Z ).lex × id id × lex lex lex Let X and Y be association schemes.The projection π : X⋋Y → X given in Proposition 2.18 induces an injection ! × Sets !→ Sets !(i.e. the pushout of {!} → I X and {!} → I Y in the category Sets of sets).