# Functoriality of Bose-Mesner algebras and profinite association schemes

Makoto Matsumoto  Mathematics Program
Graduate School of Advanced Science and Engineering
Hiroshima University, 739-8526 Japan<EMAIL_ADDRESS>,
Kento Ogawa  Mathematics Program
Graduate School of Advanced Science and Engineering
Hiroshima University, 739-8526 Japan<EMAIL_ADDRESS>and
Takayuki Okuda  Mathematics Program
Graduate School of Advanced Science and Engineering
Hiroshima University, 739-8526 Japan<EMAIL_ADDRESS>
###### Abstract.

We show that taking the set of primitive idempotents of commutative
association schemes is a functor from the category of commutative association
schemes with surjective morphisms to the category of finite sets with
surjective partial functions. We then consider projective systems of
commutative association schemes consisting of surjections (which we call
profinite association schemes), for which Bose-Mesner algebra is defined, and
describe a Delsarte theory on such schemes. This is another method for
generalizing association schemes to those on infinite sets, related with the
approach by Barg and Skriganov. Relation with $(t,m,s)$-nets and
$(t,s)$-sequences is studied. We reprove some of the results of Martin-Stinson
from this viewpoint.

###### Key words and phrases:

Association Scheme, Profinite set, Delsarte theory, $(t,m,s)$-net,
$(t,s)$-sequence, Ordered Hamming scheme, LP-program

###### 2020 Mathematics Subject Classification:

05E30 Association schemes, strongly regular graphs, 20D60 Arithmetic and
combinatorial problems, 20E18 Limits, profinite groups

The first author is partially supported by JSPS Grants-in-Aid for Scientific
Research JP18K03213. The second author is partially supported by JST SPRING,
Grant Number JPMJSP2132. The third author is partially supported by JSPS
Grants-in-Aid for Scientific Research JP20K03589, JP20K14310, and JP22H01124.

##  1\. Introduction

Association schemes are central objects in algebraic combinatorics, with many
interactions with other areas of mathematics. It is natural to try to take
projective limits of a system of association schemes, but there seems to be
some obstruction to have a natural Bose-Mesner algebra. We show that a
surjective morphism between commutative association schemes behaves well with
respect to the two product structures of Bose-Mesner algebras and primitive
idempotents, which gives rise to a notion of profinite association schemes and
their Delsarte theory. They are closely related with profinite groups. We
study kernel schemes and ordered Hamming schemes introduced by Martin-
Stinson[16] as examples. We reprove their characterization of $(t,m,s)$-net as
a design in an ordered Hamming scheme, and give a characterization of
$(t,s)$-sequences in terms of profinite association schemes.

##  2\. Surjections among association schemes and functoriality

Let ${\mathbb{N}}$ denote the set of natural numbers including $0$, and
${\mathbb{N}}_{>0}$ the set of positive integers.

###  2.1. Category of association schemes

Let us recall the notion of association schemes briefly. See [1][6] for
details. We summarize basic terminologies.

######  Definition 2.1.

Let $X$ be a set. By $\\#X$ we denote the cardinality of $X$. If $X$ has a
topology, $C(X)$ denotes the vector space of continuous functions from $X$ to
${\mathbb{C}}$. If a topology is not specified, $X$ is regarded to be a
discrete space. For $S\subset X$, we define the indicator function of $S$ as a
function $\chi_{S}:X\to\\{0,1\\}$ where $\chi(x)=1$ if and only if $x\in S$.
If $S$ is a singleton $\\{x\\}$, we denote the indicator function by
$\delta_{x}$. For a map $f:X\to Y$ between finite sets, $f^{\dagger}:C(Y)\to
C(X)$ denotes the pull back: $g\in C(Y)\mapsto g\circ f$. It is injective
(respectively surjective) if $f$ is surjective (respectively injective). By
the multiplication of functions, $C(X)$ is a unital commutative ring. This
multiplication is called the Hadamard product. For a finite set $X$, the set
$C(X\times X)$ is naturally identified with the set of complex valued square
matrices of size $\\#(X)$, and the matrix product is given by
$AB(x,z)=\sum_{y\in X}A(x,y)B(y,z)$. The Hadamard product $\circ$ is given by
the component wise product, namely, $(A\circ B)(x,z)=A(x,z)B(x,z)$. (Note that
$\circ$ may denote the composition of mappings, but no confusions would
occur.)

######  Definition 2.2.

Let $X$, $I$ be finite sets, and $R:X\times X\to I$ a surjection. We call
$(X,R,I)$ an association scheme, if the following properties are satisfied.
For each $i\in I$, $R^{-1}(i)$ may be regarded as a relation on $X$, denoted
by $R_{i}$. Let $A_{i}$ be the corresponding adjacency matrix in $C(X\times
X)$.

  1. (1)

There is an $i_{0}\in I$ such that $A_{i_{0}}$ is an identity matrix.

  2. (2)

Consider the injection $R^{\dagger}:C(I)\to C(X\times X)$. Its image $A_{X}$
is closed by the matrix product.

  3. (3)

$A_{X}$ is closed under the transpose of $C(X\times X)$.

The algebra $A_{X}$ with the two multiplications (the Hadamard product and the
matrix product) is called the Bose-Mesner algebra of $(X,R,I)$. We may use the
same notation $i_{0}$ for distinct association schemes. The number of $1$ in a
row of $A_{i}$ is called the $i$-th valency and denoted by $k_{i}$. 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 $(X,R,I)$ by an abuse of language. We follow MacLane [15] for the
terminologies of the category theory, in particular, functors, projective
systems, and limits. The association schemes form a category, by the
following[11][25].

######  Definition 2.3.

Let $(X,R,I)$, $(X^{\prime},R^{\prime},I^{\prime})$ be association schemes. A
morphism of association schemes from $X$ to $X^{\prime}$ is a pair of mappings
$f:X\to X^{\prime}$ and $g:I\to I^{\prime}$ such that the following diagram
commutes:

$\begin{array}[]{ccc}X\times X&\to&I\\\ f\times f\downarrow\phantom{f\times
f}&\circlearrowleft&\phantom{g}\downarrow g\\\ X^{\prime}\times
X^{\prime}&\to&I^{\prime}.\\\ \end{array}$ (2.1)

It is clear that the surjectivity of $f$ implies that of $g$. The morphism
$(f,g)$ is said to be surjective if $f$ is surjective.

###  2.2. Functoriality of Bose-Mesner algebra

######  Proposition 2.4.

If $(f,g):X\to X^{\prime}$ is a surjective morphism of association schemes,
then for any $x^{\prime}\in X^{\prime}$, the cardinality of the fiber
$\\#f^{-1}(x^{\prime})$ is $\\#X/\\#X^{\prime}$.

###### Proof.

The cardinality of the fiber is the summation of the valencies of $R_{i}$ for
$i$ with $g(i)=i_{0}^{\prime}$, hence independent of the choice of
$x^{\prime}$, which implies the result. This is well-known [25], and it holds
under a weaker condition (for unital regular relation partitions, see [12]). ∎

######  Theorem 2.5.

  1. (1)

Let $(f,g):(X,R,I)\to(X^{\prime},R^{\prime},I^{\prime})$ be a morphism. It
induces an injection $C(I^{\prime})\to C(I)$ and hence $\Psi:A_{X^{\prime}}\to
A_{X}$. Then $\Psi$ is a morphism of unital rings with respect to the Hadamard
product.

  2. (2)

Suppose that $f$ is surjective. We define the convolution product
$\bullet_{X}$ on $A_{X}$ by

$(A\bullet_{X}B)(x,z)=\frac{1}{\\#X}\sum_{y\in X}A(x,y)B(y,z)$

(i.e., the matrix product normalized by the factor of $\\#X$). Then, $A_{X}$
has a unit $E_{X}:=\\#XA_{i_{0}}$, and $\Psi$ is a morphism of rings with
respect to the convolution product (which may not map the unit of
$A_{X^{\prime}}$ to the unit of $A_{X}$).

  3. (3)

The vector space $C(X)$ is a left $A_{X}$-module by the following action:

$A_{X}\times C(X)\to C(X),(A,h)\mapsto(A\bullet h)$

given by

$(A\bullet h)(x)=\frac{1}{\\#X}\sum_{y\in X}A(x,y)h(y),$

and the unit $E_{X}$ acts trivially. This module structure is compatible with
$\Psi$ in the sense that the following diagram commutes:

$\begin{array}[]{ccc}A_{X}\times C(X)&\to&C(X)\\\ \Psi\times
f^{\dagger}\uparrow\phantom{f^{\dagger}\times
f^{\dagger}}&\circlearrowleft&\phantom{g\dagger}\uparrow f^{\dagger}\\\
A_{X^{\prime}}\times C(X^{\prime})&\to&C(X^{\prime}).\\\ \end{array}$ (2.2)

If we consider $C(X^{\prime})\subset C(X)$, then the element
$\Psi(E_{X^{\prime}})\in A_{X}$ acts on $C(X)$ as a projector $C(X)\to
C(X^{\prime})$.

###### Proof.

For a morphism of finite sets $X\to Y$, the induced map $C(Y)\to C(X)$ is a
morphism of unital rings with respect to the Hadamard product. The first
statement follows from this and the commutative diagram (2.1). For the second
part, take $A,B\in A_{X}^{\prime}$. (In fact, one may take $A,B\in
C(X^{\prime}\times X^{\prime})$; the following arguments depend only on the
fact that the cardinality of the fiber of $f$ is constant.) Then

$\displaystyle\Psi(A\bullet_{X^{\prime}}B)(x,z)$ $\displaystyle=$
$\displaystyle(A\bullet_{X^{\prime}}B)(f(x),f(z))$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X^{\prime}}\sum_{y^{\prime}\in
X^{\prime}}A(f(x),y^{\prime})B(y^{\prime},f(z))$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X^{\prime}}\frac{1}{\\#f^{-1}(y^{\prime})}\sum_{y\in
X}A(f(x),f(y))B(f(y),f(z))$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X^{\prime}}\frac{\\#X^{\prime}}{\\#X}\sum_{y\in
X}\Psi(A)(x,y))\Psi(B)(y,z)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in X}\Psi(A)(x,y))\Psi(B)(y,z)$
$\displaystyle=$ $\displaystyle(\Psi(A)\bullet_{X}\Psi(B))(x,z).$

For the third part, it is a routine calculation to check that $A_{X}$ acts on
$C(X)$ as a unital ring. For $A\in A_{X^{\prime}}$ and $h\in C(X^{\prime})$,
the compatibility follows from:

$\displaystyle f^{\dagger}((A\bullet h))(x)$ $\displaystyle=$
$\displaystyle(A\bullet h)(f(x))$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X^{\prime}}\sum_{y^{\prime}\in
X^{\prime}}A(f(x),y^{\prime})h(y^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X^{\prime}}\frac{1}{\\#f^{-1}(y^{\prime})}\sum_{y\in
f^{-1}(y^{\prime}),y^{\prime}\in X^{\prime}}A(f(x),f(y))h(f(y))$
$\displaystyle=$
$\displaystyle\frac{1}{\\#X^{\prime}}\frac{\\#X^{\prime}}{\\#X}\sum_{y\in
Y}\Psi(A)(x,y))(f^{\dagger}h)(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in X}\Psi(A)(x,y))(f^{\dagger}h)(y)$
$\displaystyle=$ $\displaystyle(\Psi(A)\bullet(f^{\dagger}h))(x).$

The compatibility implies that $\Psi(E_{X^{\prime}})$ trivially acts on the
image of $C(X^{\prime})$ in $C(X)$. For $A\in A_{X^{\prime}}$ and $h\in C(X)$,

$\displaystyle(\Psi(A)\bullet h))(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in X}\Psi(A)(x,y)h(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in X}A(f(x),f(y))h(y),$

which depends only on $f(x)$, and hence $\Psi(A)\bullet h$ lies in
$f^{\dagger}(C(X^{\prime}))\subset C(X)$. Hence $\Psi(E_{X^{\prime}})$ is a
projection $C(X)\to C(X^{\prime})$. ∎

######  Definition 2.6.

Let ${\operatorname{AS}}_{\operatorname{surj}}$ be the category of association
schemes with surjective morphisms.

######  Corollary 2.7.

Let ${\operatorname{Alg}}_{HC}$ (HC means Hadamard and convolution) be the
category of finite dimensional ${\mathbb{C}}$-vector spaces $A$ with:

  1. (1)

one associative multiplication (Hadamard product) which gives a commutative
unital semi-simple ${\mathbb{C}}$-algebra structure to $A$,

  2. (2)

one (possibly non-commutative) associative multiplication (convolution
product) which gives a unital semi-simple ${\mathbb{C}}$-algebra structure to
$A$.

(Note that in this case, semi-simplicity is equivalent to that $A$ is a direct
product of a finite number of matrix algebras over ${\mathbb{C}}$). Morphisms
are injective ${\mathbb{C}}$-linear maps preserving the both two products and
the unit for the Hadamard product (we don’t require preservation of unit for
the convolution product).

Then, the correspondence $X\mapsto A_{X}$ gives a contravariant functor from
${\operatorname{AS}}_{\operatorname{surj}}$ to ${\operatorname{Alg}}_{HC}$.

###### Proof.

It is proved that $A_{X}$ has two products. Semi-simplicity is well-known,
c.f. [25]. (It is enough to show that there is no nilpotent ideal, but for any
nonzero $A\in A_{X}$, the product with its unitary conjugate $A\bullet
A^{*}=\frac{1}{\\#X}AA^{*}$ is not nilpotent, hence $A_{X}$ is semi-simple.)
Functoriality is easy to check, using Theorem 2.5. ∎

######  Corollary 2.8.

Let us consider the category ${\operatorname{Mod}}$ of $R$-modules, whose
object is a pair of a unital commutative ring $R$ and an $R$-module $M$, and a
morphism from $(R,M)$ to $(R^{\prime},M^{\prime})$ is a pair of a ring
morphism $f:R\to R^{\prime}$ and a ${\mathbb{Z}}$-module morphism $g:M\to
M^{\prime}$ which makes the following diagram commute:

$\begin{array}[]{ccc}R\times M&\to&M\\\ f\times g\downarrow\phantom{f\times
g}&\circlearrowleft&\phantom{g}\downarrow g\\\ R^{\prime}\times
M^{\prime}&\to&M^{\prime}.\\\ \end{array}$

Then, the correspondence $X\mapsto(A_{X},C(X))$ is a contravariant functor
from ${\operatorname{AS}}_{\operatorname{surj}}$ to ${\operatorname{Mod}}$.

###### Proof.

The correspondence is given in Theorem 2.5. It is a contravariant functor,
since each of the correspondences $X\mapsto A_{X}$, $X\mapsto C(X)$ is a
contravariant functor. ∎

######  Remark 2.9.

In [8], French constructed a sub-category of association schemes, and a
covariant functor from it to the category of algebra. Our construction
produces a contravariant functor, which seems to be of different nature.

###  2.3. Commutative association schemes and primitive idempotents

By semisimplicity and Artin-Wedderburn Theorem, any Bose-Mesner algebra
$A_{X}$ (with convolution product) is isomorphic to a product of matrix
algebras over ${\mathbb{C}}$. We assume that $X$ is commutative. Then, $A_{X}$
is, as a unital ring, isomorphic to a direct product of copies of
${\mathbb{C}}$. Namely,
$A_{X}\cong{\mathbb{C}}\times\cdots\times{\mathbb{C}}=:{\mathbb{C}}^{n}$ for
some $n\in{\mathbb{N}}$, and this decomposition is unique as a direct product
of rings.

######  Proposition 2.10.

Let $A$ be a ring isomorphic to ${\mathbb{C}}^{n}$ (with componentwise
multiplication). An element $j\in A$ corresponding to an element of
${\mathbb{C}}^{n}$ with one coordinate $1$ and the other coordinates $0$ is
called a primitive idempotent of $A_{X}$. It is characterized by the
idempotent property $j^{2}=j$, $j\neq 0$ (this is equivalent to that the each
coordinate is $0$ or $1$ and at least one $1$ exists) and that there is no
nonzero idempotent $j^{\prime}\neq j$ such that $jj^{\prime}=j^{\prime}$ holds
(this says only one coordinate is $1$). Any idempotent is uniquely a sum of
primitive idempotents.

######  Definition 2.11.

Let ${\operatorname{FinSets}}$ be the category of finite sets. Let
${\operatorname{FinSets}}_{\operatorname{ps}}$ be the category of finite sets
and partial surjective functions.

Recall that for sets $X$ and $Y$, a partial function $f$ from $X$ to $Y$
consists of a pair of a subset ${\operatorname{dom}}(f)\subset X$ (which may
be empty) and a function
$f|_{{\operatorname{dom}}(f)}:{\operatorname{dom}}(f)\to Y$. Two partial
functions are equal if they have the same domain and the same function on the
domain. A partial function is denoted by $f:X\rightharpoonup Y$. For $S\subset
Y$, $f^{-1}(S):=\\{x\in X\mid x\in{\operatorname{dom}}(f),f(x)\in S\\}$.
Composition $g\circ f$ of partial functions $f:X\rightharpoonup Y$ and
$g:Y\rightharpoonup Z$ has domain $f^{-1}({\operatorname{dom}}(g))$. A partial
function $f$ is surjective if $\operatorname{Im}(f):=\\{f(x)\mid
x\in{\operatorname{dom}}(f)\\}\subset Y$ is $Y$.

######  Proposition 2.12.

Let ${\operatorname{Alg}}_{cs}$ be the category of rings isomorphic to
${\mathbb{C}}^{n}$ for some $n$, with injective ${\mathbb{C}}$-linear ring
morphisms which may not preserve the unit. For
$A\in{\operatorname{Alg}}_{cs}$, we define $J(A)$ as the finite set of
primitive idempotents of $A$. For a finite set $I$, we define $C(I)$ the set
of maps from $I$ to ${\mathbb{C}}$ as defined above. These are contravariant
functors, and give contra-equivalence between ${\operatorname{Alg}}_{cs}$ and
${\operatorname{FinSets}}_{\operatorname{ps}}$.

###### Proof.

Note first that the notion of $f:X\rightharpoonup Y$ is the notion of a family
of disjoint subsets $S_{y}\subset X\ \ (y\in Y)$, where $S_{y}=f^{-1}(y)$, and
the surjectivity of $f$ is equivalent to that $S_{y}\neq\emptyset$ for all
$y\in Y$. To show that $J$ is a functor, let $\varphi:A\to B$ be an injective
${\mathbb{C}}$-algebra homomorphism. For $j\in J(A)$, $\varphi(j)$ is a
nonzero idempotent. Thus, it is a non-empty sum of elements of $J(B)$, which
gives a non-empty subset $S_{j}$ of $J(B)$ by Proposition 2.10. We construct a
partial surjection $J(B)\rightharpoonup J(A)$ by: for $j\in J(A)$, elements of
$S_{j}$ is mapped to $j$. Note that if $j\neq j^{\prime}\in J(A)$, then
$jj^{\prime}=0$ and hence $S_{j}\cap S_{j^{\prime}}=\emptyset$. Since
$S_{j}\neq\emptyset$, the partial function is surjective. To check the
functoriality, we consider
$A\stackrel{{\scriptstyle\varphi}}{{\to}}B\stackrel{{\scriptstyle\phi}}{{\to}}C$.
The construction of partial surjection for $\varphi$ is done by assigning to
$j\in J(A)$ the set $S_{j}\subset J(B)$ such that summation of elements in
$S_{j}$ is $\varphi(j)$. Similarly, for each $j^{\prime}\in S_{j}$, we have
$S_{j^{\prime}}\subset J(C)$. Thus, $J(\varphi)\circ J(\phi)$ maps
$\coprod_{j^{\prime}\in S_{j}}S_{j^{\prime}}$ to $j$. This is the same with
$J(\phi\circ\varphi)$.

The functoriality of $I\to C(I)$ can be also checked in an elementary manner.
It is easy to check that the two functors give contra-equivalence. We omit the
detail. ∎

######  Definition 2.13.

Let ${\operatorname{AS}}_{\operatorname{csurj}}$ be a full subcategory of the
commutative association schemes of
${\operatorname{AS}}_{\operatorname{surj}}$, and ${\operatorname{Alg}}_{cHC}$
a full subcategory of ${\operatorname{Alg}}_{HC}$ consisting of algebras with
commutative convolution multiplication. By the remark at the beginning of
Section 2.3, we have a contravariant functor
${\operatorname{AS}}_{\operatorname{csurj}}\to{\operatorname{Alg}}_{cHC}$.

######  Corollary 2.14.

The composite of the three functors
${\operatorname{AS}}_{\operatorname{csurj}}\to{\operatorname{Alg}}_{cHC}$,
${\operatorname{Alg}}_{cHC}\to{\operatorname{Alg}}_{\operatorname{cs}}$
forgetting the Hadamard product, and
${\operatorname{Alg}}_{\operatorname{cs}}\to{\operatorname{FinSets}}_{\operatorname{ps}}$
gives a covariant functor
$AS_{\operatorname{surj}}\to{\operatorname{FinSets}}_{\operatorname{ps}}$. We
denote this composition functor by $J$. Thus, $(X,R,I)\mapsto J(X,R,I)$ is a
functor, which corresponds $X$ to the sets of primitive idempotents of $A_{X}$
with respect to (the normalized) convolution products.

######  Definition 2.15.

Let $j_{0}$ denote the element of $J(X,R,I)$, corresponding to $J_{X}$, where
$J_{X}$ denotes the matrix in $C(X\times X)$ with components all $1$. This
element acts on $C(X)$ as a projection to the space of constant functions. We
use the symbol $j_{0}$ for different association schemes, similarly to
$i_{0}$.

###  2.4. Trivial examples and remarks

In the theory of commutative association schemes, the set $I$ and the set $J$
are considered to be “dual.” Thus, it is natural to seek the relation between
$J$’s for a morphism of association schemes (while a morphism between $I$’s is
given by definition). The previous result says that there is a natural
relation, if the morphism of association schemes is surjective. We couldn’t
generalize the result for general (non-surjective) morphisms.

For a surjective morphism between association schemes, the corresponding
partial surjection may be a properly partial function, and may be not an
injection, as follows.

######  Definition 2.16.

Let $G$ be a finite group. The triple $(G,R,I)$ where $I=G$ and
$R(x,y)=x^{-1}y$ is a (possibly non-commutative) association scheme called the
thin scheme of $G$.

######  Remark 2.17.

Let $G$ be a finite abelian group, and $f:G\to H$ be a surjective homomorphism
of abelian groups. Then we have the thin scheme of $G$ and the thin scheme of
$H$, and $f$ is a surjective morphism of the association schemes. In this
case, $J$ for $G$ is naturally isomorphic to the character group $G^{\vee}$,
and one can check that the functor $J$ gives $G^{\vee}\rightharpoonup
H^{\vee}$, which comes from the natural inclusion $\iota:H^{\vee}\to G^{\vee}$
($x\in G^{\vee}$ maps to $y\in H^{\vee}$ if $\iota(y)=x$, and $x$ is outside
the domain if there is no such $y$). This gives an example that
$J(X)\rightharpoonup J(X^{\prime})$ is not a map but a partial function.

######  Remark 2.18.

For any association scheme $(X,R,I)$, the mappings $f={\operatorname{id}}_{X}$
and $g:I\to\\{0,1\\}$, $g(i_{0})=0$ and $g(i)=1$ if $i\neq i_{0}$, gives a
morphism of association schemes $(X,R,I)\to(X,g\circ R,\\{0,1\\})$. In this
case, $J(X,g\circ R,\\{0,1\\})$ consists of $j_{0}$ and $E_{X}-j_{0}$. If
$(X,R,I)$ is commutative, then the image of $j\in J(X,R,I)$ in $J(X,g\circ
R,\\{0,1\\})$ is $j_{0}$ if $j=j_{0}$, and $E_{X}-j_{0}$ if $j\neq j_{0}$.
This gives an example that $J(X)\rightharpoonup J(X^{\prime})$ is not
injective.

##  3\. Profinite association schemes

###  3.1. Projective system of association schemes

We recall the notion of projective system.

######  Definition 3.1.

(Projective system)

We work in a fixed category ${\mathcal{C}}$. Let $\Lambda$ be a directed
ordered set, namely, $\Lambda$ is non-empty and for any
$\alpha,\beta\in\Lambda$, there is a $\gamma\in\Lambda$ with
$\alpha\leq\gamma$ and $\beta\leq\gamma$. Let $X_{\lambda}(\lambda\in\Lambda)$
be a family of objects. For any $\lambda\leq\lambda^{\prime}$, a morphism
$p_{\lambda^{\prime},\lambda}:X_{\lambda^{\prime}}\to X_{\lambda}$ is
specified, and they satisfy the commutativity condition

$p_{\lambda^{\prime},\lambda}p_{\lambda^{\prime\prime},\lambda^{\prime}}=p_{\lambda^{\prime\prime},\lambda}$

for any $\lambda\leq\lambda^{\prime}\leq\lambda^{\prime\prime}$, and
$p_{\lambda,\lambda={\operatorname{id}}_{X_{\lambda}}}$. Then the system
$(X_{\lambda},p_{\lambda,\lambda^{\prime}})$ is called a projective system in
${\mathcal{C}}$. The morphisms $p_{\lambda^{\prime},\lambda}$ are called the
structure morphisms of the projective system. The dual notion (i.e. the
direction of morphism is inverted in the definition) is called an inductive
system.

Let $X_{\lambda}$ be a projective system of association schemes. Then, each of
$X_{\lambda}$, $I_{\lambda}$ is a projective system of finite sets. If all
morphisms are surjective, then by Theorem 2.5, we have an inductive system of
Bose-Mesner algebras $A_{X_{\lambda}}$, and if they are commutative, by
Corollary 2.14 a projective system
$J_{X_{\lambda}}:=J(X_{\lambda},R_{\lambda},I_{\lambda})$ (with structure
morphisms being partially surjective maps).

######  Definition 3.2.

Let $\Lambda$ be a directed ordered set. A projective system
$(X_{\lambda},R_{\lambda},I_{\lambda})$ $(\lambda\in\Lambda)$ in
${\operatorname{AS}}_{\operatorname{surj}}$ is called a profinite association
scheme. (This includes the notion of usual association scheme, as the case
where $\\#\Lambda=1$.) We define profinite sets (see Lemma 3.4)
${{{X}^{{\kern-1.04996pt}\wedge}}}:=\varprojlim X_{\lambda}$ and
${{{I}^{{\kern-1.04996pt}\wedge}}}:=\varprojlim I_{\lambda}$ with projective
limit topologies. A commutative profinite association scheme is a projective
system of commutative association schemes. In this case, we define
${{{J}^{{\kern-1.04996pt}\wedge}}}$ similarly: an element
$\hat{j}\in{{{J}^{{\kern-1.04996pt}\wedge}}}$ is the set of pair
$\lambda\in\Lambda$ and compatible family of elements $j_{\mu}\in J_{\mu}$ for
all $\mu\geq\lambda$, divided by the equivalence relation
$(\lambda,j_{\mu})\sim(\lambda^{\prime},j^{\prime}_{\mu^{\prime}})$ defined by
the existence of $\lambda^{\prime\prime}\geq\lambda$,
$\lambda^{\prime\prime}\geq\lambda^{\prime}$ such that for all
$\mu\geq\lambda^{\prime\prime}$ $j_{\mu}=j^{\prime}_{\mu}$ holds. An open
basis of ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is given by the inverse image of
$j_{\lambda}\in J_{\lambda}$ through
${{{J}^{{\kern-1.04996pt}\wedge}}}\rightharpoonup J_{\lambda}$.

######  Proposition 3.3.

The natural projections ${{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\lambda}$ and
${{{I}^{{\kern-1.04996pt}\wedge}}}\to I_{\lambda}$ are surjective, the induced
map
${{{R}^{{\kern-1.04996pt}\wedge}}}:{{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}}\to{{{I}^{{\kern-1.04996pt}\wedge}}}$
is surjective, and ${{{J}^{{\kern-1.04996pt}\wedge}}}\rightharpoonup
J_{\lambda}$ is a partial surjection.

The first two statements are well-known for projective systems of compact
Hausdorff spaces with continuous morphisms, [22, Proposition 1.1.10, Lemma
1.1.5]. We need to prove the third statement, for partial surjections. It
follows from the first statement, but for the self-containedness and for
explaining the natural (projective limit) topology, we give a proof for the
both.

######  Lemma 3.4.

  1. (1)

Let $X_{\lambda}$ be a projective system of finite sets and surjections.
Define ${{{X}^{{\kern-1.04996pt}\wedge}}}$ as $\varprojlim X_{\lambda}$. Then,
the natural map ${{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\lambda}$ is
surjective. The set ${{{X}^{{\kern-1.04996pt}\wedge}}}$ is called a profinite
set, and has a natural topology, called profinite topology, which is compact
Hausdorff. For every $x_{\lambda}\in X_{\lambda}$, its inverse image in
${{{X}^{{\kern-1.04996pt}\wedge}}}$ is clopen, and these form an open basis.

  2. (2)

Let $J_{\lambda}$ be a projective system of finite sets and partial
surjections. Define ${{{J}^{{\kern-1.04996pt}\wedge}}}$ as above. Then, the
natural partial map ${{{J}^{{\kern-1.04996pt}\wedge}}}\rightharpoonup
J_{\lambda}$ is a partial surjection. The set
${{{J}^{{\kern-1.04996pt}\wedge}}}$ has a natural topology, which is locally
compact and Hausdorff. For every $j_{\lambda}\in J_{\lambda}$, its inverse
image in ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is clopen, and these form an open
basis.

  3. (3)

Let $X_{\lambda}$, $Y_{\lambda}$ be projective systems of finite sets over the
same directed ordered set $\Lambda$, and let $f_{\lambda}:X_{\lambda}\to
Y_{\lambda}$ be a family of maps which commute with the structure morphisms.
This induces a continuous map

${{{f}^{{\kern-1.04996pt}\wedge}}}:\varprojlim X_{\lambda}\to\varprojlim
Y_{\lambda}.$

If every $f_{\lambda}$ is surjective, then ${{{f}^{{\kern-1.04996pt}\wedge}}}$
is surjective.

###### Proof.

(1): Recall the construction of the projective limit $\varprojlim X_{\lambda}$
in the category of set. It is a subset of the direct product
$\prod_{\lambda\in\Lambda}X_{\lambda}$ defined as the intersection of

$S_{\mu^{\prime},\mu}:=\\{(s_{\lambda})_{\lambda}\in\prod_{\lambda\in\Lambda}X_{\lambda}\mid
s_{\mu}=p_{\mu^{\prime},\mu}(s_{\mu^{\prime}})\\}$

for all $\mu^{\prime},\mu\in\Lambda$ with $\mu^{\prime}\geq\mu$. (This means
that $(s_{\lambda})_{\lambda}$ belongs to $\varprojlim X_{\lambda}$ if and
only if they satisfy $p_{\mu^{\prime},\mu}(s_{\mu^{\prime}})=s_{\mu}$ for
every $\mu^{\prime}\geq\mu$.) Each finite set is equipped with the discrete
topology, which is compact and Hausdorff. By Tychonoff’s theorem, the product
is compact and Hausdorff. Because each component is Hausdorff,
$S_{\mu^{\prime},\mu}$ is a closed subset, and the intersection $\varprojlim
X_{\lambda}$ is compact and Hausdorff. Take any $x_{\delta}\in X_{\delta}$. We
shall show that this is in the image from $\varprojlim X_{\lambda}$. For each
$\alpha\in\Lambda$, $\alpha\geq\delta$, we consider the closed set

$T_{\alpha}:=\\{(s_{\lambda})_{\lambda}\in\prod_{\lambda\in\Lambda}X_{\lambda}\mid
s_{\delta}=x_{\delta},s_{\beta}=p_{\alpha,\beta}(s_{\alpha})\mbox{ for all
}\beta\leq\alpha\\}.$

There is an element $y_{\alpha}\in X_{\alpha}$ such that
$p_{\alpha,\delta}(y_{\alpha})=x_{\delta}$, since $p$ is surjective. Then,
there is a system $p_{\alpha,\beta}(y_{\alpha})$ which is an element of
$T_{\alpha}$, and thus $T_{\alpha}$ is nonempty. Now we take the intersection
of $T_{\alpha}$ for all $\alpha\geq\delta$. For any finite number of
$\alpha_{i}$, we may take $\gamma$ as an upper bound of all $\alpha_{i}$. Then
the finite intersection contains $T_{\gamma}$, which is nonempty. By
compactness, the intersection of $T_{\alpha}$ for $\alpha\geq\delta$ is
nonempty. Take an element $(g_{\lambda})_{\lambda}$ in the intersection. It
lies in $\varprojlim X_{\lambda}$, and it projects to $x_{\delta}$ via
$\varprojlim X_{\lambda}\to X_{\delta}$. From the definition of the direct
product topology, the inverse image of $x_{\lambda}\in X_{\lambda}$ for
various $\lambda$ forms an open basis. They are clopen, since $x\in
X_{\lambda}$ is clopen.

(2): Choose an element $\star$ which is contained in no $J_{\lambda}$’s. Let
$J_{\lambda}^{\star}$ be $J_{\lambda}\cup\\{\star\\}$. For a partial
surjection $p:J_{\lambda}\rightharpoonup J_{\mu}$, we associate a surjection
$p^{\star}:J_{\lambda}^{\star}\to J_{\mu}^{\star}$ as follows. For
$x\in{\operatorname{dom}}(p)$, $p^{\star}(x)=p(x)$. For
$x\notin{\operatorname{dom}}(p)$, $p^{\star}(x)=\star$. In particular,
$p^{\star}(\star)=\star$. Let us take the projective limit as in the previous
case to obtain ${{{J}^{{\kern-1.04996pt}\wedge}}}^{\star}:=\varprojlim
J_{\lambda}^{\star}$. Then for any $\lambda$, by (1), we have a surjection

$p^{\star}_{\lambda}:{{{J}^{{\kern-1.04996pt}\wedge}}}^{\star}\to
J_{\lambda}^{\star}.$

It is clear that there is a unique element
$\widehat{\star}\in{{{J}^{{\kern-1.04996pt}\wedge}}}^{\star}$ that is mapped
to $\star$ for every $J_{\lambda}^{\star}$. We define
${{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}:=(\varprojlim
J_{\lambda}^{\star})\setminus\\{\widehat{\star}\\}$, and a partial surjection
$p_{\lambda}:{{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}\rightharpoonup
J_{\lambda}$ by: for $x\in{{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}$, $x$ is
outside of the domain of $p_{\lambda}$ if $p^{\star}_{\lambda}(x)=\star$, and
otherwise $p_{\lambda}(x)=p^{\star}_{\lambda}(x)$. It is easy to check that
$p_{\lambda}$ is a partial surjection (using the surjectivity of
$p^{\star}_{\lambda}$), and $p_{\lambda}$ for various $\lambda$ is compatible
with the structure morphisms $p_{\mu^{\prime},\mu}$.

Since ${{{J}^{{\kern-1.04996pt}\wedge}}}^{\star}$ is compact Hausdorff, its
open subset ${{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}$ is locally compact
Hausdorff, and has an open basis as in the case (1). We need to prove that
${{{J}^{{\kern-1.04996pt}\wedge}}}$ in Definition 3.2 is canonically
isomorphic to ${{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}$. Let $\tilde{J}$ be
the set of pairs $(\lambda,(j_{\mu})_{\mu})$ where $(j_{\mu})_{\mu}$ is a
compatible system for $\mu\geq\lambda$. For such a pair, we assign an element
of ${{{J}^{{\kern-1.04996pt}\wedge}}}^{\star}$ as follows. For any
$\alpha\in\Lambda$, take any $\beta$ with $\beta\geq\alpha$ and
$\beta\geq\lambda$. Define $j_{\alpha}$ as
$p^{\star}_{\beta,\alpha}(j_{\mu})$. This is well-defined by the commutativity
of the structure morphisms, and gives a map
$f:\tilde{J}\to{{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}$. This is
surjective. Two pairs $(\lambda,(j_{\mu})_{\mu})$,
$(\lambda^{\prime},(j^{\prime}_{\mu^{\prime}})_{\mu^{\prime}})$ have the same
image if and only if there is a $\lambda^{\prime\prime}$ with
$\lambda^{\prime\prime}\geq\lambda$,
$\lambda^{\prime\prime}\geq\lambda^{\prime}$ such that for any
$\mu\geq\lambda^{\prime\prime}$, $j_{\mu}=j^{\prime}_{\mu}$ holds. This gives
a bijection
${{{J}^{{\kern-1.04996pt}\wedge}}}:=\tilde{J}/\sim\to{{{J}^{{\kern-1.04996pt}\wedge}}}^{\prime}$.

(3): See [22, Lemma 1.1.5]. ∎

######  Remark 3.5.

The above construction gives a category equivalence between the category of
sets with partial maps and the category of sets with one base point. Using
this, it is not difficult to show that ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is
the projective limit in the category of sets with partial surjections.

Later it will be proved that, for any commutative profinite association
scheme, ${{{J}^{{\kern-1.04996pt}\wedge}}}$ has the discrete topology
(Proposition 3.10).

######  Definition 3.6.

For a profinite association scheme $(X_{\lambda},R_{\lambda},I_{\lambda})$,
its Bose-Mesner algebra is defined by
$A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}:=\varinjlim A_{X_{\lambda}}$. For a
commutative profinite association scheme, through the natural isomorphism
$C(J_{\lambda})\to A_{X_{\lambda}}$, we have an isomorphism $\varinjlim
C(J_{\lambda})\to A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$ which maps the point-
wise product to the convolution product, and an isomorphism $\varinjlim
C(I_{\lambda})\to A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$ which maps the point-
wise product to the Hadamard product.

Let
$C({{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}})$
denote the set of complex valued continuous functions on
${{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}}$
(and similarly $C({{{I}^{{\kern-1.04996pt}\wedge}}})$ be the set of continuous
functions on ${{{I}^{{\kern-1.04996pt}\wedge}}}$). Since we have inductive
families of injections

$A_{X_{\lambda}}\to C(X_{\lambda}\times X_{\lambda})\to
C({{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}}),$

by universality, we have a canonical injective morphism
$A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}\to
C({{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}})$.
similarly, we have an injection $\varinjlim C(I_{\lambda})\to
C({{{I}^{{\kern-1.04996pt}\wedge}}})$.

######  Definition 3.7.

For $j_{\lambda}\in J_{\lambda}$, we denote by $E_{j_{\lambda}}\in
A_{X_{\lambda}}$ the primitive idempotent $j_{\lambda}$ (the notation
introduced to distinguish an element of $J$ with an element of
$A_{X_{\lambda}}$). It is an element of $C(X_{\lambda}\times X_{\lambda})$,
and acts on $C(X_{\lambda})$ by the normalized convolution defined in Theorem
2.5. Let $C(X_{\lambda})_{j_{\lambda}}$ be the image of $E_{j_{\lambda}}$ by
this action.

######  Proposition 3.8.

Take $j_{\lambda}\in J_{\lambda}$. For $\mu\geq\lambda$, we may regard
$C(X_{\lambda})\subset C(X_{\mu})$. Then $\Psi(E_{j_{\lambda}})$ given in
Theorem 2.5 acts on $g\in C(X_{\mu})$ by $\Psi(E_{j_{\lambda}})\bullet g$. The
operator $\Psi(E_{j_{\lambda}})$ gives a splitting $C(X_{\mu})\to
C(X_{\lambda})_{j_{\lambda}}$ to $C(X_{\lambda})_{j_{\lambda}}\subset
C(X_{\mu})$.

###### Proof.

By Theorem 2.5, $\Psi(E_{X_{\lambda}})\in{{\operatorname{End}}}(C(X_{\mu}))$
is a projection to $C(X_{\lambda})$. Then

$\Psi(E_{j_{\lambda}})=\Psi(E_{j_{\lambda}}\bullet
E_{X_{\lambda}})=\Psi(E_{j_{\lambda}})\bullet\Psi(E_{X_{\lambda}})$

is a projection $C(X_{\mu})\to C(X_{\lambda})\to
C(X_{\lambda})_{j_{\lambda}}$. ∎

######  Definition 3.9.

Let $j$ be an element of ${{{J}^{{\kern-1.04996pt}\wedge}}}$. Then $j$ is said
to be isolated at $\lambda$ to $j_{\lambda}$, if there is a $j_{\lambda}\in
J_{\lambda}$ such that the inverse image of $j_{\lambda}$ is the singleton
$\\{j\\}$.

######  Proposition 3.10.

The cardinality of the inverse image of $j_{\lambda}$ in $J_{\mu}$ for
$\mu\geq\lambda$ does not exceed the dimension of
$C(X_{\lambda})_{j_{\lambda}}$, which is finite. Consequently, for each
$j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$, there exists a $\lambda$ at which $j$
is isolated. Thus, ${{{J}^{{\kern-1.04996pt}\wedge}}}$ has the discrete
topology.

###### Proof.

Suppose that the inverse image of $j_{\lambda}$ in $J_{\mu}$ is
$\\{j_{1},j_{2},\dots,j_{n}\\}$. Then the operator
$\Psi(E_{j_{\lambda}})=\sum_{i=1}^{n}\Psi(E_{j_{i}})$ on $C(X_{\mu})$ is a
projection to $C(X_{\lambda})_{j_{\lambda}}$. Thus $C(X_{\mu})_{j_{i}}$
$(i=1,\ldots,n)$ are nonzero direct summands of
$C(X_{\lambda})_{j_{\lambda}}$, and hence $n$ does not exceed the dimension of
$C(X_{\lambda})_{j_{\lambda}}$. This means that for any $\mu\geq\lambda$, the
cardinality of the inverse image of $j_{\lambda}$ is bounded, and hence there
is a $\mu^{\prime}$ such that the cardinality is constant for all
$\mu\geq\mu^{\prime}$. Let us fix $j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$.
Take any $\lambda_{0}$ such that $j$ is in the domain of the partial map to
$J_{\lambda_{0}}$. We put $j_{\lambda_{0}}$ to the image of $j$. Then by the
arguments above, we can find a $\lambda$ such that the cardinality of the
inverse image of $j_{\lambda_{0}}$ in $J_{\lambda^{\prime}}$ is constant for
all $\lambda^{\prime}\geq\lambda$. We put $j_{\lambda}$ to the image of $j$ in
$J_{\lambda}$. Then $j_{\lambda}$ is in the inverse image of $j_{\lambda_{0}}$
and the cardinality of the inverse image of $j_{\lambda}$ in
$J_{\lambda^{\prime}}$ should be one for any $\lambda^{\prime}\geq\lambda$.
This implies that the inverse image of $j_{\lambda}$ in
${{{J}^{{\kern-1.04996pt}\wedge}}}$ is just $\\{j\\}$, namely, $j$ is isolated
at $\lambda$. We have proved that $\\{j\\}$ is a clopen subset of
${{{J}^{{\kern-1.04996pt}\wedge}}}$ for any
$j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$, and hence
${{{J}^{{\kern-1.04996pt}\wedge}}}$ has the discrete topology. ∎

######  Corollary 3.11.

We have $\varinjlim
C(J_{\lambda})=\bigoplus_{j\in{{{J}^{{\kern-0.74997pt}\wedge}}}}{\mathbb{C}}\cdot
j$. We write $E_{j}$ for the image of $j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$
in $A_{{{X}^{{\kern-0.74997pt}\wedge}}}$. Then
$\bigoplus_{j\in{{{J}^{{\kern-0.74997pt}\wedge}}}}{\mathbb{C}}\cdot
E_{j}=A_{{{X}^{{\kern-0.74997pt}\wedge}}}$. We may use the notation
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$ for $\varinjlim C(J_{\lambda})$,
since it is the set of functions (automatically continuous) on
${{{J}^{{\kern-1.04996pt}\wedge}}}$ with finite ($=$compact) support. Through
the isomorphism $C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})\to
A_{{{X}^{{\kern-0.74997pt}\wedge}}}$, $E_{j}\in
A_{{{X}^{{\kern-0.74997pt}\wedge}}}$ corresponds to the indicator function
$\delta_{j}\in C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$.

######  Remark 3.12.

In the Pontryagin duality, the dual of a compact abelian group is a discrete
group. Since ${{{I}^{{\kern-1.04996pt}\wedge}}}$ is profinite and compact, the
above discreteness of ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is an analogue to
this fact.

We summarize the results of this subsection, to compare with the case of
finite commutative association schemes.

######  Theorem 3.13.

Let $X_{\lambda}$ be a commutative profinite association scheme. Notations are
as in Definition 3.6 and Corollary 3.11. We have the following commutative
diagram, where all the vertical arrows are injections:

$\begin{array}[]{ccccccccc}&&C({{{J}^{{\kern-1.04996pt}\wedge}}})&&C({{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}})&\leftarrow&C({{{I}^{{\kern-1.04996pt}\wedge}}})&&\\\
&&\uparrow&&\uparrow&&\uparrow&&\\\
C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})&=&\varinjlim
C(J_{\lambda})&\simeq&A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}&\simeq&\varinjlim
C(I_{\lambda})&=&C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}}),\\\ \end{array}$
(3.1)

where:

  1. (1)

The set ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is discrete. The image of the left
vertical arrow is the set of compact support functions
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$ (i.e. takes 0 except but finite
points in ${{{J}^{{\kern-1.04996pt}\wedge}}}$).

  2. (2)

The image of the right vertical arrow is the set of locally constant functions
$C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$, which is dense with respect to
the supremum norm, see the next lemma.

######  Lemma 3.14.

Let ${{{X}^{{\kern-1.04996pt}\wedge}}}$ be the projective limit of a
projective system of finite sets $X_{\lambda}$ for a directed ordered set
$\Lambda$.

  1. (1)

Let $C$ be a compact subspace of ${{{X}^{{\kern-1.04996pt}\wedge}}}$, and
$O_{a}$ $(a\in A)$ be an open covering of $C$. Then, there exists a
$\mu\in\Lambda$ such that by putting $S$ to the image of $C$ in $X_{\mu}$, the
inverse image of $s\in S$ in ${{{X}^{{\kern-1.04996pt}\wedge}}}$ constitutes a
(finite and disjoint) open covering of $C$, and each of these open sets is
contained in one of $O_{a}$.

  2. (2)

The image of $\varinjlim C(X_{\lambda})\to
C({{{X}^{{\kern-1.04996pt}\wedge}}})$ is the set of locally constant functions
on ${{{X}^{{\kern-1.04996pt}\wedge}}}$, and is dense with respect to the
supremum norm.

###### Proof.

(1): For $x_{\lambda}\in X_{\lambda}$, we denote by $B_{x_{\lambda}}$ the
inverse image of $x_{\lambda}$ in ${{{X}^{{\kern-1.04996pt}\wedge}}}$, which
we call a $\lambda$-ball. This gives an open basis, and thus every $O_{a}$ is
a union of $\lambda$-balls for various $\lambda$. Since $C$ is compact and
covered by these balls, we may find a finite number of these balls which
covers $C$ and each contained in some $O_{a}$. We take an upper bound $\mu$
for these finite number of $\lambda$. Let $S$ be the image of $C$ in
$X_{\mu}$. Then, $B_{s_{\mu}}(s_{\mu}\in S)$ are finite, clopen, and mutually
disjoint balls covering $C$, such that each ball is contained in some $O_{a}$.

(2) Let $f$ be a locally constant function. Then, for any
$x\in{{{X}^{{\kern-1.04996pt}\wedge}}}$, there is an open neighborhood $O_{x}$
so that $f$ is constant on $O_{x}$. These constitute an open covering of
${{{X}^{{\kern-1.04996pt}\wedge}}}$. Applying (1) for
$C={{{X}^{{\kern-1.04996pt}\wedge}}}$ to find $\mu$ with $S=X_{\mu}$, where
each $\mu$-ball is contained in some $O_{x}$, and hence $f$ is constant on
each ball. This means that $f$ is an image of an element in $C(X_{\mu})$.
Conversely, any function in $C(X_{\mu})$ clearly maps to a locally constant
function on ${{{X}^{{\kern-1.04996pt}\wedge}}}$.

We shall show the density. For any $f\in C({{{X}^{{\kern-1.04996pt}\wedge}}})$
and $\epsilon>0$, by continuity, for all
$x\in{{{X}^{{\kern-1.04996pt}\wedge}}}$ we have an open neighborhood $O_{x}$
such that $y\in O_{x}$ implies $|f(x)-f(y)|<\epsilon$. By applying (1) for
$C={{{X}^{{\kern-1.04996pt}\wedge}}}$ to find a $\mu$ with $S=X_{\mu}$. We see
that the $\mu$-balls cover ${{{X}^{{\kern-1.04996pt}\wedge}}}$, and for each
$x_{\mu}\in X_{\mu}$, we define $g(x_{\mu})=f(x^{\prime})$ by choosing any
$x^{\prime}$ in $B_{x_{\mu}}$. Denote the projection by
${\operatorname{pr}}:{{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\mu}$. Then for
any $y\in{{{X}^{{\kern-1.04996pt}\wedge}}}$,
$g({\operatorname{pr}}(y))=f(x^{\prime})$ for some $x^{\prime}\in
B_{{\operatorname{pr}}(y)}$. Since $B_{{\operatorname{pr}}(y)}\subset O_{x}$
for some $x$, the values of $f$ in this ball differ at most by $2\epsilon$,
namely,

$|g({\operatorname{pr}}(y))-f(y)|=|f(x^{\prime})-f(y)|<2\epsilon.$

This shows that

$||{\operatorname{pr}}^{\dagger}(g)-f||_{\sup}<2\epsilon,$

which is the desired density. ∎

###  3.2. Inner product and orthogonality

######  Proposition 3.15.

For a profinite association scheme, ${{{X}^{{\kern-1.04996pt}\wedge}}}$ has a
natural probability regular Borel measure $\mu$, whose push-forward is the
normalized (probability) counting measure on $X_{\lambda}$ for each $\lambda$.
If ${{{X}^{{\kern-1.04996pt}\wedge}}}$ is a profinite group $G$ considered as
a projective system of $G/N_{\lambda}$ for (finite index) open normal
subgroups, then the above $\mu$ coincides with the probability Haar measure.

###### Proof.

This is a direct consequence of Choksi [5] and a general property of measures
(see Halmos [10]), as follows. Each $X_{\lambda}$ is finite discrete (hence
compact Hausdorff) with normalized counting measure, and the measure is
preserved by the uniformness of the cardinality of each fiber as in
Proposition 2.4. This means that the system is an inverse family of measured
spaces [5, Definition 2], and hence ${{{X}^{{\kern-1.04996pt}\wedge}}}$ has a
weak measure structure: a set $M$ of ring of sets and a function $\mu$ from
$M$ to non-negative real numbers, such that
${{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\lambda}$ preserves the measure. In
this case, $M$ is the set of open sets, and
$\mu({{{X}^{{\kern-1.04996pt}\wedge}}})=1$. Then [5, Theorem 2.2] tells that
this $\mu$ on ${{{X}^{{\kern-1.04996pt}\wedge}}}$ is extended to a measure on
$\sigma$-ring generated by $M$, namely, the Borel sets. There it is also
proved that for any measurable set $E$ with finite measure and any
$\epsilon>0$, there is a compact set $C\subset E$ with
$\mu(E)<\mu(C)+\epsilon$. This proves the inner regularity of $\mu$ ([10, §52,
P.224]). By taking the complement, for any $\epsilon>0$ and measurable set
$E^{c}$, there is an open $U\supset E^{c}$ with
$1-\mu(E^{c})<1-\mu(U)+\epsilon$, i.e., $\mu(E^{c})>\mu(U)-\epsilon$, which
proves the outer regularity of $\mu$, and hence $\mu$ is regular.

Let $\nu$ be the probability Haar measure of $G$. Then, for any (finite index)
open normal subgroup $N$, $\nu(N)$ is $\frac{1}{\\#(G/N)}$, since $\\#(G/N)$
cosets of $N$ disjointly covers $G$. This gives an inverse family of measured
spaces, and the above construction gives a regular Borel measure $\mu$. The
Haar measure is regular [10, §58, Theorem B]. For the coset $gN$,
$\mu(gN)=\nu(gN)$. The cosets $gN$ for $g$ and $N$ varying make an open basis.
By the outer regularity, for any compact set $E$ and any $\epsilon>0$, there
is an open set $O\supset E$ with $\mu(E)>\mu(O)-\epsilon$. Use Lemma 3.14 for
$C=E$ and the open covering $O$, to find $\mu$, $S\subset X_{\mu}$. Then
$B_{s_{\mu}}(s_{\mu}\in S)$ constitutes an open cover ${\mathcal{C}}$ of $E$
consisting of a finite number of disjoint cosets (contained in $O$) such that
the sum of the measure of each coset in ${\mathcal{C}}$, denoted by
$\mu({\mathcal{C}})$ $(\mu({\mathcal{C}})\leq\mu(O))$, satisfying

$\mu({\mathcal{C}})\geq\mu(E)\geq\mu({\mathcal{C}})-\epsilon,$ (3.2)

but if the same argument simultaneously applied for both $\mu$ and $\nu$ for
the $O$ satisfying both $\mu(E)>\mu(O)-\epsilon$ and $\nu(E)>\nu(O)-\epsilon$,
we may take the same ${\mathcal{C}}$ for the both and the same inequality
(3.2) holds for $\nu$, and because consisting of a finite number of disjoint
cosets, $\nu({\mathcal{C}})=\mu({\mathcal{C}})$ holds, which implies
$\mu(E)=\nu(E)$. This means that the restrictions of $\nu$ and $\mu$ to
compact sets give the same regular content, which implies $\nu=\mu$ by [10,
§54, Theorem B]. ∎

######  Proposition 3.16.

For a finite set $X$, we define a (normalized) Hermitian inner product on
$C(X)$ by

$(f,g):=\frac{1}{\\#X}\sum_{x\in X}f(x)\overline{g(x)}.$

For a profinite association scheme, it is defined on
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$ by

$(f,g):=\int_{x\in{{{X}^{{\kern-0.74997pt}\wedge}}}}f(x)\overline{g(x)}d\mu(x).$

Then $C(X_{\lambda})\to C({{{X}^{{\kern-1.04996pt}\wedge}}})$ preserves the
inner products.

This is clear since ${{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\lambda}$
preserves the measures.

######  Definition 3.17.

Let $V,W$ be ${\mathbb{C}}$-vector spaces with Hermitian inner products. For a
linear morphism $A:V\to W$, its conjugate $A^{*}:W\to V$ (which may not
exists) is characterized by the property

$(Af,g)_{W}=(f,A^{*}g)_{V}$

for any $f\in V$, $g\in W$. It is unique if exists. An endomorphism $A:W\to W$
is said to be Hermitian if $A=A^{*}$, as usual.

Let $W$ be again a ${\mathbb{C}}$-vector space with Hermitian inner product,
and $V$ is a subspace with restricted inner product. Let $V^{\perp}$ be the
space of vectors which is orthogonal to any vector in $V$. If $V\oplus
V^{\perp}$ is equal to $W$, then $V$ is said to be an orthogonal component of
$W$.

######  Definition 3.18.

Let $W$ be a ${\mathbb{C}}$-linear space with Hermitian inner product. For a
linear subspace $V\subset W$, let $p:W\to W$ be a projection to $V$, i.e., the
image of $p$ is $V$ and $p|_{V}={\operatorname{id}}_{V}$. Then by linear
algebra $W$ is decomposed as the direct sum, $W=V\oplus\operatorname{Ker}p$.
We say that $p$ is an orthogonal projector to $V$ if
$V\oplus\operatorname{Ker}p$ is an orthogonal direct sum. An orthogonal
projector to $V$ exists uniquely, if $V$ is an orthogonal component.

######  Proposition 3.19.

Let $W$ be a ${\mathbb{C}}$-linear space with Hermitian inner product,
$V\subset W$ be an orthogonal component with orthogonal projector $p$, and
$\iota:V\hookrightarrow W$ be the inclusion. We denote by $q:W\to V$ the
restriction of the codomain of $p$. Then we have $\iota^{*}=q$ and
$q^{*}=\iota$. For $A:V\to V$, if $A^{*}$ exists, then

$(\iota\circ A\circ q)^{*}=\iota\circ A^{*}\circ q:V\to V.$ (3.3)

We sometimes denote $\iota\circ A\circ q$ simply by $A$. We make a remark when
this abuse of notation is used.

###### Proof.

For $w\in W$ and $v\in V$, write $w=w_{V}+w_{V}^{\prime}$ with $w_{V}\in V$
and $w_{V}^{\prime}\in V^{\perp}$. Then

$(qw,v)_{V}=(w_{V},v)_{V}=(\iota w_{V},\iota
v)_{W}=(w_{V}+w_{V}^{\prime},\iota v)_{W}=(w,\iota v)_{W},$

which shows that $\iota^{*}=q$ and $q^{*}=\iota$. Thus (3.3) follows. ∎

######  Proposition 3.20.

Let $W$ be a ${\mathbb{C}}$-vector space with Hermitian inner product, and
$p:W\to W$ an idempotent, namely, $p^{2}=p$. By linear algebra,
$W=\operatorname{Im}(p)\oplus\operatorname{Ker}(p)$ and $p$ is a projector to
$V:=\operatorname{Im}(p)$. Then, $p$ is an orthogonal projector if and only if
$p=p^{*}$. If $A\in{{\operatorname{End}}}(V)$ is an orthogonal projector to
$U\subset V$, then $\iota Aq:W\rightarrow W$ is a projector to $U\subset W$.

###### Proof.

If $p^{2}=p$, then $W=V\oplus\operatorname{Ker}p$ for
$V=\operatorname{Im}(p)$. Thus any element $w,w^{\prime}\in W$ is written as
$v+k,v^{\prime}+k^{\prime}$, respectively. If $V\oplus\operatorname{Ker}p$ is
orthogonal,

$(p(v+k),v^{\prime}+k^{\prime})=(pv,v^{\prime}+k^{\prime})=(v,v^{\prime})=(v+k,pv^{\prime})=(v+k,p(v^{\prime}+k^{\prime})),$

which shows $p=p^{*}$. For the converse, assume $p=p^{*}$. Then for $v\in V$,
$k\in\operatorname{Ker}p$, $(v,k)=(pv,k)=(v,pk)=(v,0)=0$, and hence $W$ and
$\operatorname{Ker}p$ are mutually orthogonal. For the last, (3.3) implies
that $\iota Aq$ is an Hermitian operator, and by
$q\circ\iota={\operatorname{id}}_{V}$ is an idempotent, whose image is $U$,
hence is an orthogonal projector to $U$. ∎

######  Definition 3.21.

Let $(X_{\lambda})$ be a profinite association scheme, fix $\lambda\in\Lambda$
and $x_{\lambda}\in X_{\lambda}$. For the surjection
${\operatorname{pr}}:{{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\lambda}$, the
fiber ${\operatorname{pr}}^{-1}(x_{\lambda})$ at $x_{\lambda}$ is a clopen
subset of ${{{X}^{{\kern-1.04996pt}\wedge}}}$. Thus, we have the restriction
of the measure $\mu$ on ${{{X}^{{\kern-1.04996pt}\wedge}}}$ to
${\operatorname{pr}}^{-1}(x_{\lambda})$, denoted by the same symbol
$({\operatorname{pr}}^{-1}(x_{\lambda}),\mu)$. Note that the volume of the
fiber ${\operatorname{pr}}^{-1}(x_{\lambda})$ is $1/\\#X_{\lambda}$.

Since ${{{X}^{{\kern-1.04996pt}\wedge}}}=\coprod_{x_{\lambda}\in
X_{\lambda}}{\operatorname{pr}}^{-1}(x_{\lambda})$, the following proposition
holds.

######  Proposition 3.22.

For any $\lambda$ and $f\in C({{{X}^{{\kern-1.04996pt}\wedge}}})$, we have

$\int_{x\in{{{X}^{{\kern-0.74997pt}\wedge}}}}f(x)d\mu(x)=\sum_{x_{\lambda}\in
X_{\lambda}}\int_{y\in{\operatorname{pr}}^{-1}(x_{\lambda})}f(y)d\mu(y).$

######  Proposition 3.23.

Let $(X_{\lambda})$ be a profinite association scheme. Then, by the surjection
${\operatorname{pr}}:{{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{\lambda}$,
$C(X_{\lambda})$ is identified with a subspace of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$ with the inner product given by
restriction. There is an orthogonal projection
$q:C({{{X}^{{\kern-1.04996pt}\wedge}}})\to C(X_{\lambda})$ given by averaging
over each fiber: for $f\in C({{{X}^{{\kern-1.04996pt}\wedge}}})$ define

$q(f)(x_{\lambda})=\frac{1}{\mu({\operatorname{pr}}^{-1}(x_{\lambda}))}\int_{x\in{\operatorname{pr}}^{-1}(x_{\lambda})}f(x)d\mu(x),$

where $\mu({\operatorname{pr}}^{-1}(x_{\lambda}))=\frac{1}{\\#X_{\lambda}}$.
Thus, $C(X_{\lambda})$ is an orthogonal component of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$.

###### Proof.

The preservation of the inner products is shown in Proposition 3.16. For any
$g\in C(X_{\lambda})$ its image is $g\circ{\operatorname{pr}}\in
C({{{X}^{{\kern-1.04996pt}\wedge}}})$ and we have

$\displaystyle q(g\circ{\operatorname{pr}})(x_{\lambda})$ $\displaystyle=$
$\displaystyle\frac{1}{\mu({\operatorname{pr}}^{-1}(x_{\lambda}))}\int_{y\in{\operatorname{pr}}^{-1}(x_{\lambda})}g({\operatorname{pr}}(y))d\mu(y)$
$\displaystyle=$ $\displaystyle
g(x_{\lambda})\frac{1}{\mu({\operatorname{pr}}^{-1}(x_{\lambda}))}\int_{y\in{\operatorname{pr}}^{-1}(x_{\lambda})}1\cdot
d\mu(y)$ $\displaystyle=$ $\displaystyle g(x_{\lambda}).$

This implies that $q$ is a projection to $C(X_{\lambda})$.

To show that $q$ is an orthogonal projection, we take
$f\in\operatorname{Ker}q$. This means that the average of $f$ on each fiber of
${\operatorname{pr}}$ is zero. To show that $q$ is orthogonal, it suffices to
show that $(f,g\circ{\operatorname{pr}})_{{{X}^{{\kern-0.74997pt}\wedge}}}=0$
for $g\in C(X_{\lambda})$ by Definition 3.18, but by applying Proposition
3.22, we have

$\displaystyle(f,g\circ{\operatorname{pr}})_{{{X}^{{\kern-0.74997pt}\wedge}}}$
$\displaystyle=$
$\displaystyle\int_{x\in{{{X}^{{\kern-0.74997pt}\wedge}}}}f(x)\overline{g({\operatorname{pr}}(x))}d\mu(x)$
$\displaystyle=$ $\displaystyle\sum_{x^{\prime}\in
X_{\lambda}}\int_{x\in{\operatorname{pr}}^{-1}(x^{\prime})}f(x)\overline{g({\operatorname{pr}}(x))}d\mu(x)$
$\displaystyle=$ $\displaystyle\sum_{x^{\prime}\in
X_{\lambda}}\overline{g(x^{\prime})}\int_{x\in{\operatorname{pr}}^{-1}(x^{\prime})}f(x)d\mu(x)$
$\displaystyle=$ $\displaystyle\sum_{x^{\prime}\in
X_{\lambda}}\overline{g(x^{\prime})}\cdot 0=0.$

∎

######  Proposition 3.24.

  1. (1)

The (convolution) algebra $A_{{{X}^{{\kern-0.74997pt}\wedge}}}$ acts on
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$, which makes
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$ an
$A_{{{X}^{{\kern-0.74997pt}\wedge}}}$-module. This is compatible with the
action of $A_{X_{\lambda}}$ on $C(X_{\lambda})$, in the sense of Corollary
2.8.

  2. (2)

The convolution unit $E_{X_{\lambda}}\in A_{X_{\lambda}}\subset
A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$ acts as an orthogonal projector to
$C(X_{\lambda})$.

  3. (3)

Let $j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$. Take a $\lambda$ where $j$ is
isolated to $j_{\lambda}$, and consider $E_{j_{\lambda}}\in A_{X_{\lambda}}$.
By the abuse of notation in Proposition 3.19, $E_{j_{\lambda}}$ extends to the
orthogonal projector

$E_{j}:C({{{X}^{{\kern-1.04996pt}\wedge}}})\to
C({{{X}^{{\kern-1.04996pt}\wedge}}})$

that projects to the domain of the composition

$C(X_{\lambda})_{j_{\lambda}}\to C(X_{\lambda})\to
C({{{X}^{{\kern-1.04996pt}\wedge}}}).$

We define

$C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}:=C(X_{\lambda})_{j_{\lambda}}.$

This is independent of the choice of the $\lambda$.

  4. (4)

For $j\neq j^{\prime}$, $E_{j}\bullet E_{j^{\prime}}=0$.

  5. (5)

We have an orthogonal decomposition

$C(X_{\lambda})=\bigoplus_{j\in
J^{\prime}}C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$

for the domain $J^{\prime}$ of the partial map from
${{{J}^{{\kern-1.04996pt}\wedge}}}$ to $J_{\lambda}$. Note that $J^{\prime}$
is finite.

###### Proof.

The action is given by: $A\in A_{X_{\lambda}}$ acts on $f\in
C({{{X}^{{\kern-1.04996pt}\wedge}}})$ by

$(A\bullet
f)(x):=\int_{y\in{{{X}^{{\kern-0.74997pt}\wedge}}}}A({\operatorname{pr}}_{\lambda}(x),{\operatorname{pr}}_{\lambda}(y))f(y)d\mu(y).$

By Proposition 3.22, this is the same with

$\sum_{y^{\prime}\in
X_{\lambda}}A({\operatorname{pr}}_{\lambda}(x),y^{\prime})(\int_{y\in{\operatorname{pr}}_{\lambda}^{-1}(y^{\prime})}f(y)d\mu(y)).$
(3.4)

Comparing with $q$ in Proposition 3.23, we have

$\displaystyle(A\bullet_{{{{X}^{{\kern-0.74997pt}\wedge}}}}f)(x)$
$\displaystyle=$ $\displaystyle\sum_{y^{\prime}\in
X_{\lambda}}A({\operatorname{pr}}_{\lambda}(x),y^{\prime})(\int_{y\in{\operatorname{pr}}_{\lambda}^{-1}(y^{\prime})}f(y)d\mu(y))$
$\displaystyle=$ $\displaystyle\sum_{y^{\prime}\in
X_{\lambda}}A({\operatorname{pr}}_{\lambda}(x),y^{\prime})\frac{1}{\\#X_{\lambda}}q(f)(y^{\prime})$
$\displaystyle=$
$\displaystyle(A\bullet_{X_{\lambda}}q(f))({\operatorname{pr}}_{\lambda}(x)),$

which shows that

$A=\iota Aq,$ (3.5)

where $\iota:C(X_{\lambda})\to C({{{X}^{{\kern-1.04996pt}\wedge}}})$ is the
injection ${\operatorname{pr}}_{\lambda}^{\dagger}$ and $A$ in the right hand
side is an endomorphism of $C(X_{\lambda})$. This and
$q\circ\iota={\operatorname{id}}_{C(X_{\lambda})}$ show that
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$ is an
$A_{{{X}^{{\kern-0.74997pt}\wedge}}}$-module. The well-definedness
(independence of the choice of $\lambda$) follows by taking a sufficiently
large $\mu$.

By (3.5) and $E_{X_{\lambda}}={\operatorname{id}}_{C(X_{\lambda})}$ on
$C(X_{\lambda})$ show that

$E_{X_{\lambda}}=\iota\circ q,$

which is the orthogonal projection to $C(X_{\lambda})$.

For $j\in J$, take a $\lambda$ where $j$ is isolated to $j_{\lambda}\in
J_{\lambda}$. By (3.5) we have

$E_{j_{\lambda}}=\iota E_{j_{\lambda}}q,$

where $E_{j_{\lambda}}$ in the right hand side is an endomorphism of
$C(X_{\lambda})$. Hence to prove that $E_{j_{\lambda}}$ is an orthogonal
projector $C({{{X}^{{\kern-1.04996pt}\wedge}}})\to
C(X_{\lambda})_{j_{\lambda}}$, it suffices to show that $E_{j_{\lambda}}\in
A_{X_{\lambda}}$ is an orthogonal projector $C(X_{\lambda})\to
C(X_{\lambda})_{j_{\lambda}}$ by Proposition 3.20. The orthogonality of
$E_{j_{\lambda}}\in A_{X_{\lambda}}$ is well known [1]. (Since
$A_{X_{\lambda}}$ is closed under the anti-isomorphism $*$ of ring,
$E_{j_{\lambda}}^{*}$ must be a primitive idempotent, and since
$E_{j_{\lambda}}E_{j_{\lambda}}^{*}\neq 0$, they must coincide, hence
$E_{j_{\lambda}}$ is Hermitian and an orthogonal projection by Proposition
3.20.) We define $E_{j}:=E_{j_{\lambda}}$, where

$E_{j}:C({{{X}^{{\kern-1.04996pt}\wedge}}})\to
C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}:=C(X_{\lambda})_{j_{\lambda}}.$

For the well-definedness, we take $\lambda$, $\lambda^{\prime}$ where $j$ is
isolated. There exists a $\mu$ with $\mu\geq\lambda$,
$\mu\geq\lambda^{\prime}$. Then $j$ is isolated to $j_{\mu}$ at $\mu$, and by
(3.5) it is not difficult to see that the $E_{j}$ defined using $\mu$ is the
same as those defined using $\lambda,\lambda^{\prime}$. For $E_{j}\bullet
E_{j^{\prime}}=0$, take a $\lambda$ where both $j$, $j^{\prime}$ are isolated
to $j_{\lambda}$, $j^{\prime}_{\lambda}$, respectively. Then, $j_{\lambda}\neq
j_{\lambda}^{\prime}$, $E_{j_{\lambda}},E_{j^{\prime}_{\lambda}}\in
A_{X_{\lambda}}$, and $E_{j_{\lambda}}\bullet E_{j^{\prime}_{\lambda}}=0$.
Thus, the relation holds in the inductive limit
$A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$.

For fixed $\lambda$, it is well-known [1] that we have an orthogonal
decomposition

$C(X_{\lambda})=\bigoplus_{j_{\lambda}\in
J_{\lambda}}C(X_{\lambda})_{j_{\lambda}}.$

(This follows from the orthogonality of $E_{j_{\lambda}}$.) Since each
$C(X_{\lambda})_{j_{\lambda}}$ is an orthogonal direct sum of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$ over the finite number of
$j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$ whose image is $j_{\lambda}$, the
claim follows. ∎

The following is an analogue to Peter-Weyl Theorem.

######  Theorem 3.25.

We have an orthogonal direct decomposition

$\varinjlim C(X_{\lambda})=\bigoplus_{j\in
J}C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}.$

Hence the right hand side is dense in $C({{{X}^{{\kern-1.04996pt}\wedge}}})$
with respect to the supremum norm.

###### Proof.

Proposition 3.24 implies that $C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$ are
mutually orthogonal, and their orthogonal sum is a subspace of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$. For any $\lambda$, $C(X_{\lambda})$ is
an orthogonal sum of a finite number of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$. Thus $\varinjlim C(X_{\lambda})$
is the direct sum of $C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$ for
$j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$. The density follows from Lemma 3.14.
∎

###  3.3. Measures and Fourier analysis

Since ${{{X}^{{\kern-1.04996pt}\wedge}}}$ has a natural probability measure,
${{{I}^{{\kern-1.04996pt}\wedge}}}$ has a pushforward measure via
${{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}}\to{{{I}^{{\kern-1.04996pt}\wedge}}}$.
It will be shown that ${{{J}^{{\kern-1.04996pt}\wedge}}}$ has a natural
measure, and these measures are closely related to the classical notions of
the multiplicity $m_{j_{\lambda}}$ for $J_{\lambda}$, and the valency
$k_{i_{\lambda}}$ for $I_{\lambda}$ (see [1][6]).

######  Definition 3.26.

Let $\mu_{{{I}^{{\kern-0.74997pt}\wedge}}}$ be the pushforward measure on
${{{I}^{{\kern-1.04996pt}\wedge}}}$ from
${{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}}$.
This is characterized as follows. For $i_{\lambda}\in I_{\lambda}$, let
${\operatorname{pr}}_{I_{\lambda}}^{-1}(i)\subset{{{I}^{{\kern-1.04996pt}\wedge}}}$
be the open set. Then

$\mu_{{{I}^{{\kern-0.74997pt}\wedge}}}({\operatorname{pr}}_{I_{\lambda}}^{-1}(i_{\lambda}))=\mu_{{{{X}^{{\kern-0.74997pt}\wedge}}}\times{{{X}^{{\kern-0.74997pt}\wedge}}}}(R_{\lambda}\circ{\operatorname{pr}}_{X_{\lambda}\times
X_{\lambda}})^{-1}(i_{\lambda}).$

######  Proposition 3.27.

In the above definition,

$\mu_{{{I}^{{\kern-0.74997pt}\wedge}}}({\operatorname{pr}}_{I_{\lambda}}^{-1}(i_{\lambda}))=\frac{k_{i_{\lambda}}}{\\#X_{\lambda}}$

holds, where $k_{i_{\lambda}}$ is the valency of the adjacency matrix
$A_{i_{\lambda}}\in A_{X_{\lambda}}$.

###### Proof.

Recall that $A_{i_{\lambda}}$ is the indicator function of
$R_{\lambda}^{-1}(i_{\lambda})$, and the sum of the rows is an integer
$k_{i_{\lambda}}$ called the valency [1]. Thus, the volume
$\mu_{{{{I}^{{\kern-0.74997pt}\wedge}}}}({\operatorname{pr}}_{I_{\lambda}}^{-1}(i_{\lambda}))$
is the volume of
$\mu_{{{{X}^{{\kern-0.74997pt}\wedge}}}}({\operatorname{pr}}_{X_{\lambda}\times
X_{\lambda}})^{-1}R_{\lambda}^{-1}(i_{\lambda})$, which is

$\displaystyle\int_{x,y\in{{{X}^{{\kern-0.74997pt}\wedge}}}\times{{{X}^{{\kern-0.74997pt}\wedge}}}}A_{i_{\lambda}}({\operatorname{pr}}_{X_{\lambda}}(x),{\operatorname{pr}}_{X_{\lambda}}(y))d\mu(x)d\mu(y)$
$\displaystyle=$
$\displaystyle\frac{1}{\\#X_{\lambda}^{2}}\sum_{x^{\prime},y^{\prime}\in
X_{\lambda}\times X_{\lambda}}A_{i_{\lambda}}(x^{\prime},y^{\prime})$
$\displaystyle=$ $\displaystyle\frac{k_{i_{\lambda}}}{\\#X_{\lambda}}.$

∎

######  Definition 3.28.

The algebra $A_{X_{\lambda}}$ is closed under the adjoint ∗. We define the
Hilbert-Schmidt inner product, which is Hermitian, as follows. For $A,B\in
A_{X_{\lambda}}$,

$(A,B)_{{\operatorname{HS}}}:=\sum_{x\in X_{\lambda}}(A\bullet B^{*})(x,x).$

For $A,B\in A_{{{X}^{{\kern-0.74997pt}\wedge}}}$, we define its Hilbert-
Schmidt inner product by

$(A,B)_{{\operatorname{HS}}}:=\int_{x\in{{{X}^{{\kern-0.74997pt}\wedge}}}}(A\bullet
B^{*})(x,x)d\mu(x).$

Then inclusion $A_{X_{\lambda}}\to A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$
preserves the inner products.

###### Proof.

For $B\in A_{X_{\lambda}}$, $B^{*}\in A_{X_{\lambda}}$ follows because of the
definition of association schemes, and the Hilbert-Schmidt inner product is
defined. For $B\in A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$, we must show that
$B^{*}$ exists in $A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$ and the integration
converges. However, $B\in A_{X_{\lambda}}$ for some $\lambda$, and $B^{*}\in
A_{X_{\lambda}}$. Since $C(X_{\lambda})$ is an orthogonal component of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$, $B^{*}=\iota B^{*}q$ is the adjoint as
an operator on $C({{{X}^{{\kern-1.04996pt}\wedge}}})$, by the abuse of
notation in Proposition 3.19. The notation $(A\bullet B^{*})(x,x)$ comes from
the middle vertical injection in (3.1). Since $A,B^{*}\in
A_{X_{\lambda}}\subset C(X_{\lambda}\times X_{\lambda})$,

$\int_{x\in{{{X}^{{\kern-0.74997pt}\wedge}}}}(A\bullet
B^{*})(x,x)d\mu(x)=\sum_{x^{\prime}\in X_{\lambda}}(A\bullet
B^{*})(x^{\prime},x^{\prime}),$

which is finite, satisfying the axioms of Hermitian inner product, and
compatible with $A_{X_{\lambda}}\to A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$. ∎

######  Proposition 3.29.

Through the natural isomorphism

$\varinjlim C(I_{\lambda})\to A_{{{{X}^{{\kern-0.74997pt}\wedge}}}},$

we obtain Hermitian inner product on $\varinjlim C(I_{\lambda})$, which is
denoted by $(-,-)_{{\operatorname{HS}}}$. Then for $f,g\in\varinjlim
C(I_{\lambda})$, we have

$(f,g)_{{\operatorname{HS}}}=\int_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}f(i)\overline{g(i)}d\mu_{{{I}^{{\kern-0.74997pt}\wedge}}}(i).$

###### Proof.

We may assume $f,g\in C(I_{\lambda})$ for some $\lambda$. Then by bilinearity
it suffices to check the equality for $f=\delta_{i_{\lambda}}$,
$g=\delta_{i_{\lambda}^{\prime}}$. Then, the right hand side is
$\delta_{i_{\lambda},i_{\lambda}^{\prime}}k_{i_{\lambda}}/\\#X_{\lambda}$ by
Proposition 3.27. In $A_{{{X}^{{\kern-0.74997pt}\wedge}}}$, these functions
correspond to $f\mapsto A_{i_{\lambda}},g\mapsto A_{i_{\lambda}^{\prime}}$,
and their inner product is

$\displaystyle\int_{x,y\in{{{X}^{{\kern-0.74997pt}\wedge}}}\times{{{X}^{{\kern-0.74997pt}\wedge}}}}A_{i_{\lambda}}({\operatorname{pr}}_{X_{\lambda}}(x),{\operatorname{pr}}_{X_{\lambda}}(y))A_{i_{\lambda}^{\prime}}({\operatorname{pr}}_{X_{\lambda}}(y),{\operatorname{pr}}_{X_{\lambda}}(x))^{*}d\mu(x)d\mu(y)$
$\displaystyle=$
$\displaystyle\frac{1}{\\#X_{\lambda}^{2}}\sum_{x^{\prime},y^{\prime}\in
X_{\lambda}\times
X_{\lambda}}A_{i_{\lambda}}(x^{\prime},y^{\prime})A_{i_{\lambda}^{\prime}}(x^{\prime},y^{\prime})$
$\displaystyle=$
$\displaystyle\delta_{i_{\lambda},i_{\lambda}^{\prime}}\frac{k_{i_{\lambda}}}{\\#X_{\lambda}}.$

This uses a well-known orthogonality of $A_{i_{\lambda}}$: for any $x\in
X_{\lambda}$ and $i_{1},i_{2}\in I_{\lambda}$, the $(x,x)$-component of the
matrix product $A_{i_{1}}\cdot^{t}A_{i_{2}}$ is the number of $y\in
X_{\lambda}$ with $R(x,y)=i_{1}$ and $R(x,y)=i_{2}$, which is zero unless
$i_{1}=i_{2}$, and in the equal case $k_{i_{1}}$. ∎

######  Lemma 3.30.

For a commutative association scheme $(X,R,I)$, we defined $J$ as the set of
primitive idempotents $E_{j}$ with respect to the product $\bullet$ in Theorem
2.5. We put a positive measure $m_{j}$ for $j\in J$, so that the inner product
with respect to $m_{j}$ in $C(J)$ is isometric to that of $A_{X}$ with respect
to ${{\operatorname{HS}}}$ inner product. Then, $m_{j}$ is the multiplicity of
$j$, defined as $\dim(C(X)_{j})$ [1].

###### Proof.

Let $E_{j},E_{j^{\prime}}\in A_{X}$ be two primitive idempotents. The
computation

$\displaystyle(E_{j},E_{j^{\prime}})_{{{\operatorname{HS}}}}=\sum_{x\in
X}E_{j}\bullet
E_{j^{\prime}}^{*}(x,x)=\delta_{jj^{\prime}}{{\operatorname{tr}}}(E_{j})=\delta_{jj^{\prime}}m_{j}$

implies that if we define $\mu_{J}(j):=m_{j}$, then the corresponding inner
product is compatible with $C(J)\to A_{X}$ with respect to the
${{\operatorname{HS}}}$-inner product on $A$. ∎

######  Proposition 3.31.

For a projective association scheme $(X_{\lambda},R_{\lambda},I_{\lambda})$,
we define a measure $\mu_{{{J}^{{\kern-0.74997pt}\wedge}}}$ on the discrete
set ${{{J}^{{\kern-1.04996pt}\wedge}}}$ by
$\mu_{{{J}^{{\kern-0.74997pt}\wedge}}}(j)=\dim
C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$. Then, the inner product on
$\varinjlim C(J_{\lambda})$ defined by

$(f,g)_{{\operatorname{HS}}}=\sum_{j\in{{{J}^{{\kern-0.74997pt}\wedge}}}}f(j)\overline{g(j)}\mu_{{{J}^{{\kern-0.74997pt}\wedge}}}(j)$

makes $\varinjlim C(J_{\lambda})\cong A_{{{X}^{{\kern-0.74997pt}\wedge}}}$
isometric with respect to ${{\operatorname{HS}}}$ inner product.

###### Proof.

Any $f,g\in\varinjlim C(J_{\lambda})$ are contained in some $C(J_{\lambda})$,
where any $j$ in the support of $f$ or $g$ is isolated at $\lambda$. The
functions $f,g$ are linear combinations of orthogonal basis $E_{j}$ of
$C(J_{\lambda})$, namely, $f=\sum_{j}f(j)E_{j}$ and $g=\sum_{j}g(j)E_{j}$ in
$A_{X_{\lambda}}$. The previous lemma says that
$(E_{j},E_{j^{\prime}})_{{\operatorname{HS}}}=\delta_{jj^{\prime}}\dim
C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$, which proves the proposition. ∎

In sum, we have the following form of Fourier transform.

######  Theorem 3.32.

The isomorphism of vector spaces $\varinjlim C(J_{\lambda})\cong
A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}\cong\varinjlim C(I_{\lambda})$ is an
isometry between the space of compact support functions
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$ with inner product defined in
Proposition 3.31 and the space of locally constant functions
$C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$ with inner product defined in
Proposition 3.29, which maps the componentwise product in
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$ to the convolution product in
$C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$.

######  Remark 3.33.

For a finite association scheme $(X,R,I)$, both $E_{j}$ $(j\in J)$ and $A_{i}$
$(i\in I)$ are orthogonal bases of $A_{X}$ with respect to
${{\operatorname{HS}}}$-inner product. The isometry

$C(I)\to C(J)$

is given by

$\displaystyle A_{i}$ $\displaystyle\mapsto$ $\displaystyle\sum_{j\in
J}p_{i}(j)E_{j}.$

The isometry shows that

$(A_{i},A_{i^{\prime}})_{{\operatorname{HS}}}=\delta_{ii^{\prime}}\mu_{I}(i)=\delta_{ii^{\prime}}\frac{k_{i}}{\\#X}$

is equal to

$\sum_{j\in J}p_{i}(j)\overline{p_{i^{\prime}}(j)}m_{j},$

which is one of the two orthogonalities of eigenmatrices as classically known
[1]. (Here the factor $\\#X$ appears in a different manner because of the
normalization in Theorem 2.5. Our $E_{j}$ is $\\#X$ times the classical
primitive idempotent, which makes $p_{i}(j)=\frac{1}{\\#X}P_{i}(j)$ where
$P_{i}(j)$ are the components of the standard eigen matrices.)

The next is an analogue of the Fourier transform for $L^{2}$ spaces.

######  Proposition 3.34.

The isometry $\varinjlim C(J_{\lambda})\to\varinjlim C(I_{\lambda})$ extends
to a unique isometry

$L^{2}({{{J}^{{\kern-1.04996pt}\wedge}}},\mu_{{{J}^{{\kern-0.74997pt}\wedge}}})\to
L^{2}({{{I}^{{\kern-1.04996pt}\wedge}}},\mu_{{{I}^{{\kern-0.74997pt}\wedge}}}).$

###### Proof.

We consider $L^{2}$-norm for all the spaces. The map $\varinjlim
C(J_{\lambda})\to L^{2}({{{J}^{{\kern-1.04996pt}\wedge}}})$ is isometric,
injective, and has a dense image, by considering the orthogonal basis $E_{j}$.
Since $L^{2}({{{J}^{{\kern-1.04996pt}\wedge}}})$ is complete, this is a
completion of $\varinjlim C(J_{\lambda})$. The map $\varinjlim
C(I_{\lambda})\to L^{2}({{{I}^{{\kern-1.04996pt}\wedge}}})$ is also isometric,
hence injective, and has a dense image (by Lemma 3.14 it is dense with respect
to supremum norm in $C({{{I}^{{\kern-1.04996pt}\wedge}}})$, and since
${{{I}^{{\kern-1.04996pt}\wedge}}}$ is compact, dense with respect to the
$L^{2}$-norm in $C({{{I}^{{\kern-1.04996pt}\wedge}}})$, and thus dense in
$L^{2}({{{I}^{{\kern-1.04996pt}\wedge}}})$), so
$L^{2}({{{I}^{{\kern-1.04996pt}\wedge}}})$ is a completion of $\varinjlim
C(I_{\lambda})$. By the universality of the completion, they are isometric. ∎

##  4\. Some examples and relation with Barg-Skriganov theory

The results of this paper are closely related to the study by Barg and
Skriganov [2]. They defined the notion of association schemes on a set $X$
with a $\sigma$-additive measure. See [2, Definition 1], where $I$ is denoted
by $\Upsilon$. Association scheme structure is given by a surjection

$R:X\times X\to I,$

with axioms similar to the finite case. The inverse image $R^{-1}(i)\subset
X\times X$ of $i\in I$ is required to be measurable in $X\times X$, and
intersection numbers $p_{ij}^{k}$ are defined as the measure of the set

$\\{z\in X\mid R(x,z)=i,R(z,y)=j\\}$

where $R(x,y)=k$, which is required to be independent of the choice of $x,y$.
The set $I$ is required to be at most countably infinite. This countability
assumption seems to be necessary, for example to avoid the case where
$p_{ij}^{k}$ is all zero for any $i,j,k$.

Let $\chi_{i}(x,y)$ be the indicator function of $R^{-1}(i)$. They define
adjacency algebras (Bose-Mesner algebras in our terminology) in [2, Section 3]
as a set of finite linear combination of $\chi_{i}(x,y)$ with complex
coefficient, with convolution product given by the integration

$(a*b)(x,y):=\int_{X}a(x,z)b(z,y)d\mu(z).$

However, it is not clear whether the adjacency algebra is closed under the
convolution product. In fact, the authors remarked in the last of [2, Section
3.1] “our arguments in this part are of somewhat heuristic nature. We make
them fully rigorous for the case of association schemes on zero-dimensional
groups; see Section 8.” Later in Section 8, they define the adjacency algebra
with an aid of locally compact zero-dimensional abelian groups and their
duality. Some of the treated objects (compact cases) are examples of profinite
association schemes in this paper.

###  4.1. Profinite groups

Let $G$ be a profinite group, that is, a projective limit of finite groups
$G_{\lambda}$ with projective limit topology. Absolute Galois groups,
arithmetic fundamental groups and $p$-adic integers are some examples appear
in arithmetic geometry [22]. There are significant amounts of studies on
analysis on abelian profinite groups, see for example [14].

####  4.1.1. Schurian profinite association schemes

We describe Schurian association schemes, and then profinite analogues. When
we say an action of a group $G$, it means the left action, unless otherwise
stated.

######  Lemma 4.1.

Let $X$ be a topological space and $G$ a group acting on $X$ such that every
element of $G$ induces a homeomorphism on $X$. Then the space of continuous
functions $C(X)$ is a left $G$-module by $f(x)\mapsto f(g^{-1}x)$. Let
$G\backslash X$ denote the quotient space with quotient topology. Let

$C(X)^{G}\subset C(X)$

denote the subspace of functions invariant by the action of $G$. There is a
canonical injection

$C(G\backslash X)\hookrightarrow C(X)$

associated with the continuous surjection $X\to G\backslash X$. Then, there is
a canonical isomorphism of ${\mathbb{C}}$-algebras

$C(X)^{G}\to C(G\backslash X),$

and hence we identify them.

###### Proof.

An element $f$ of $C(X)^{G}$ is a continuous function which takes one same
value for each $G$-orbit of $X$. Thus, it gives a mapping $G\backslash
X\to{\mathbb{C}}$. By the universality of the quotient topology, this is
continuous, hence lies in $C(G\backslash X)$. Conversely, a function in
$C(G\backslash X)$ gives a continuous function in $C(X)$, which is invariant
by $G$. ∎

######  Proposition 4.2.

(Schurian association scheme)

Let $G$ be a finite group, $H$ a subgroup, and let $X:=G/H$ be the quotient.
Then, we have a natural bijection

$G\backslash(X\times X)\to H\backslash G/H,$ (4.1)

where $G$ acts diagonally. We define

$I:=G\backslash(X\times X).$

The quotient mapping

$R:X\times X\to I$

is an association scheme, which is called a Schurian scheme.

We give a proof, since descriptions used in the proof are necessary for the
profinite case. The equality (4.1) is given by a surjection

$(g_{1}H,g_{2}H)\mapsto Hg_{1}^{-1}g_{2}H,$

which certainly factors through

$G\backslash(g_{1}H,g_{2}H)\mapsto Hg_{1}^{-1}g_{2}H,$

and the converse map is given by

$HgH\mapsto(H,gH),$

where the check of the well-definedness is left to the reader.

######  Lemma 4.3.

Let $C(X)^{\vee}:=\operatorname{Hom}_{\mathbb{C}}(C(X),{\mathbb{C}})$ be the
dual space, which is regarded as a left $G$-module, by letting $g\in G$ act on
$\xi\in C(X)^{\vee}$ by $g(\xi)f=\xi(g^{-1}(f))$. Then we have an isomorphism
of $G$-modules

$C(X)\stackrel{{\scriptstyle\vee}}{{\to}}C(X)^{\vee},\quad\delta_{x}\mapsto{\operatorname{ev}}_{x}$

for every $x\in X$, where $\delta_{x}$ is the indicator function of $x$ and
${\operatorname{ev}}_{x}$ is the evaluation at $x$.

###### Proof.

It is well-known that $\\{\delta_{x}\mid x\in X\\}$ is a linear basis of
$C(X)$, and $\\{{\operatorname{ev}}_{x}\mid x\in X\\}$ is a linear basis of
$C(X)^{\vee}$. We compute

$g(\delta_{x})(y)=\delta_{x}(g^{-1}y)=\delta_{gx}(y).$

On the other hand,

$g({\operatorname{ev}}_{x})(f)={\operatorname{ev}}_{x}(g^{-1}(f))=(g^{-1}(f))(x)=f(gx)={\operatorname{ev}}_{gx}(f).$

∎

######  Proposition 4.4.

The identification of a matrix and a linear map is obtained by an isomorphism
of $G$-modules

$C(X\times X)=C(X)\otimes
C(X)\stackrel{{\scriptstyle{\operatorname{id}}\otimes\vee}}{{\to}}C(X)\otimes
C(X)^{\vee}\cong\operatorname{Hom}_{\mathbb{C}}(C(X),C(X)).$ (4.2)

###### Proof.

An element $A\in C(X\times X)$ is mapped as

$\displaystyle A$ $\displaystyle\mapsto$ $\displaystyle\sum_{x,y\in
X^{2}}A(x,y)\delta_{x}\otimes\delta_{y}$ $\displaystyle\mapsto$
$\displaystyle\sum_{x,y\in
X^{2}}A(x,y)\delta_{x}\otimes{\operatorname{ev}}_{y}.$

The last element is regarded as an element of
$\operatorname{Hom}_{\mathbb{C}}(C(X),C(X))$ by mapping $f=\sum_{y\in
X}f(y)\delta_{y}\in C(X)$ to

$(\sum_{x,y\in
X^{2}}A(x,y)\delta_{x}\otimes{\operatorname{ev}}_{y})(f)=\sum_{x\in
X}\sum_{y\in Y}A(x,y)f(y)\delta_{x},$

which is the multiplication of a matrix to a vector. ∎

######  Proposition 4.5.

By taking the $G$-invariant part of (4.2), we have

$\operatorname{Hom}_{G}(C(X),C(X))\cong C(X\times X)^{G}=C(G\backslash(X\times
X))=C(H\backslash G/H),$ (4.3)

where the left $\operatorname{Hom}_{G}$ denotes the endomorphism as
$G$-modules.

###### Proof.

This follows from (4.1), Lemma 4.1 and the definition of $G$-homomorphisms. ∎

######  Corollary 4.6.

The construction in Proposition 4.2 gives an association scheme. The
endomorphism ring $\operatorname{Hom}_{G}(C(X),C(X))$ is isomorphic to the
Bose-Mesner algebra $A_{X}$.

###### Proof.

By the construction of $I=G\backslash(X\times X)$, the set of the adjacency
matrices is closed under the transpose and contains the identity matrix. Their
linear span is closed by the matrix product, since the adjacency matrices form
a linear basis of (4.3) and the matrix product corresponds to the composition
of the endomorphism ring $\operatorname{Hom}_{G}(C(X),C(X))$. This gives an
association scheme with Bose-Mesner algebra
$\operatorname{Hom}_{G}(C(X),C(X))$. ∎

######  Definition 4.7.

Let $G$ be a finite group acting on a ${\mathbb{C}}$-linear space $V$. By the
representation of finite groups, $V$ is decomposed into irreducible
representations. For a fixed irreducible representation $W$, the multiplicity
of $W$ in $V$ is the dimension of the vector space

$\operatorname{Hom}_{G}(W,V).$

We say $V$ is multiplicity free, if the multiplicity is at most one for any
irreducible representation $W$.

######  Corollary 4.8.

The Schurian association scheme is commutative, if and only if $C(X)$ is
multiplicity free. In this case, the set of the primitive idempotents is
canonically identified with the set of the irreducible representations of $G$
in $C(X)$.

###### Proof.

If the multiplicity is one for each irreducible representation, Schur’s lemma
implies that $\operatorname{Hom}_{G}(C(X),C(X))$ is a direct sum of copies of
${\mathbb{C}}$ (each copy corresponding to each irreducible representation in
$C(X)$) as a ring, and hence is commutative. If there is an irreducible
component with multiplicity $m\geq 2$, $\operatorname{Hom}_{G}(C(X),C(X))$
contains a matrix algebra $M_{m}({\mathbb{C}})$, and thus is not commutative.
∎

See [22] for the following definition and properties (c.f. Section 3.1 above).

######  Definition 4.9.

Let $G_{\lambda}(\lambda\in\Lambda)$ be a projective system of surjections of
finite groups. A profinite group is a topological group which is isomorphic to

${{{G}^{{\kern-1.04996pt}\wedge}}}:=\varprojlim G_{\lambda},$

where each $G_{\lambda}$ has the discrete topology. Because the projection
${\operatorname{pr}}_{\lambda}:{{{G}^{{\kern-1.04996pt}\wedge}}}\to
G_{\lambda}$ is surjective and continuous, we identify $G_{\lambda}$ with
${{{G}^{{\kern-1.04996pt}\wedge}}}/N_{\lambda}$, where $N_{\lambda}$ is the
kernel of the ${\operatorname{pr}}_{\lambda}$, which is a clopen normal
subgroup of ${{{G}^{{\kern-1.04996pt}\wedge}}}$, since $G_{\lambda}$ is finite
and discrete. By Tychonoff’s theorem, ${{{G}^{{\kern-1.04996pt}\wedge}}}$ is a
compact Hausdorff group, with a basis of open neighborhood of the unit element
$e$ consisting of $N_{\lambda}$. Being Hausdorff, the intersection of
$N_{\lambda}$ is $\\{e\\}$.

We prepare a well-known lemma for describing profinite Schurian association
schemes.

######  Lemma 4.10.

Let ${{{G}^{{\kern-1.04996pt}\wedge}}}=\varprojlim
G_{\lambda}=\varprojlim{{{G}^{{\kern-1.04996pt}\wedge}}}/N_{\lambda}$ be a
profinite group. Let $H$ and $K$ be closed subgroups of
${{{G}^{{\kern-1.04996pt}\wedge}}}$. Give a quotient topology on
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ by

$q:{{{G}^{{\kern-1.04996pt}\wedge}}}\to
H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K.$

  1. (1)

The image of $K$ in $G_{\lambda}$ is $KN_{\lambda}$ (which is a union of
cosets of $N_{\lambda}$, and hence a subset of $G_{\lambda}$). The same
statement holds for $H$.

  2. (2)

We have the following commutative diagram of continuous surjections, where
$\alpha$ is a bijection.

$\begin{array}[]{ccc}H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K&&\\\
\downarrow&\searrow&\\\
HN_{\lambda}\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/KN_{\lambda}&\stackrel{{\scriptstyle\alpha}}{{\to}}&HN_{\lambda}\backslash
G_{\lambda}/KN_{\lambda}.\end{array}$ (4.4)

  3. (3)

The $q:{{{G}^{{\kern-1.04996pt}\wedge}}}\to
H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is an open morphism, and
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is compact Hausdorff.

  4. (4)

The finite sets $HN_{\lambda}\backslash G_{\lambda}/KN_{\lambda}$ constitute a
projective system, with a canonical homeomorphism

$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K\stackrel{{\scriptstyle\sim}}{{\to}}\varprojlim
HN_{\lambda}\backslash G_{\lambda}/KN_{\lambda}.$

###### Proof.

(1) When we consider $G_{\lambda}$ as a quotient of
${{{G}^{{\kern-1.04996pt}\wedge}}}$, $G_{\lambda}$ is a set of cosets
$gN_{\lambda}$. The image of $K$ is the union of $kN_{\lambda}$ for $k\in K$,
which is nothing but $KN_{\lambda}$.

(2) Every set in the diagram is a quotient of
${{{G}^{{\kern-1.04996pt}\wedge}}}$ with various equivalence relation. The
vertical arrow is obviously well-defined. The mapping $\alpha$ is bijective,
since as a quotient of ${{{G}^{{\kern-1.04996pt}\wedge}}}$, they are the same.
The slanting arrow is merely the composition of the two. We recall that
${{{G}^{{\kern-1.04996pt}\wedge}}}\to
H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is a quotient map. Thus, every
map in the diagram is continuous, since the maps from
${{{G}^{{\kern-1.04996pt}\wedge}}}$ are continuous (where the mappings to the
two sets at the bottom of the diagram factors through
${{{G}^{{\kern-1.04996pt}\wedge}}}/N_{\lambda}$, which is finite and
discrete).

(3) The open sets $sN_{\lambda}$ for $s\in{{{G}^{{\kern-1.04996pt}\wedge}}}$
and various $\lambda$ constitute an open basis of
${{{G}^{{\kern-1.04996pt}\wedge}}}$. The image of $sN_{\lambda}$ in
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is $HsKN_{\lambda}$ (since
$N_{\lambda}$ is normal), whose inverse image in
${{{G}^{{\kern-1.04996pt}\wedge}}}$ is again $HsKN_{\lambda}$, which is open.
Thus, by the definition of quotient topology, $HsKN_{\lambda}$ is open in
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$, and hence $q$ is open.

Take two distinct elements $HgK$, $Hg^{\prime}K$ in
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$. These are compact disjoint
subsets of the Hausdorff space ${{{G}^{{\kern-1.04996pt}\wedge}}}$. It is well
known that there are an open set
$O_{a}\subset{{{G}^{{\kern-1.04996pt}\wedge}}}$ containing $HgK$ and an open
set $O_{b}\supset Hg^{\prime}K$ that separate $HgK$ and $Hg^{\prime}K$. By (1)
of Lemma 3.14 for the union of $HgK$ and $Hg^{\prime}K$, we may take $\mu$
such that open sets of the form $sN_{\mu}\subset O_{a}$ with $sN_{\mu}\cap
HgK\neq\emptyset$ cover $HgK$ and open sets $sN_{\mu}\subset O_{b}$ with
$sN_{\mu}\cap Hg^{\prime}K\neq\emptyset$ cover $Hg^{\prime}K$, that is,
$HgKN_{\mu}\subset O_{a}$ and $Hg^{\prime}KN_{\mu}\subset O_{b}$ are disjoint.
Since $q$ is open, $HgKN_{\mu}$ and $Hg^{\prime}KN_{\mu}$ are open sets in
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ which have no intersection
and are neighborhoods of $HgK$, $Hg^{\prime}K$ (which are considered as
elements of $H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$), respectively.
Thus $H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is Hausdorff.

(4) If we see these sets as quotients of ${{{G}^{{\kern-1.04996pt}\wedge}}}$
as in (4.4), it is clear that these form a projective system. By the
universality of projective limit, we have a continuous map

$f:H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K\to\varprojlim
HN_{\lambda}\backslash G_{\lambda}/KN_{\lambda}$

by (2). The image of $f$ is dense, since for any element $\xi$ in the
projective limit and its open neighborhood, there is a smaller open
neighborhood which has the form
${\operatorname{pr}}_{\lambda}^{-1}{\operatorname{pr}}_{\lambda}(\xi)$ for
some $\lambda$, since these form a basis of open neighborhoods of $\xi$.
Because

$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K\to HN_{\lambda}\backslash
G_{\lambda}/KN_{\lambda}$

is surjective, the density of the image follows. Since
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is compact (being a quotient
of a compact set ${{{G}^{{\kern-1.04996pt}\wedge}}}$), the image is compact
and closed, hence $f$ is surjective. Since
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$ is Hausdorff, any two
distinct points $HgK$, $Hg^{\prime}K$ have disjoint open neighborhoods
$HgKN_{\lambda}$, $Hg^{\prime}KN_{\lambda}$, respectively, as proved above,
hence their images in $HN_{\lambda}\backslash G_{\lambda}/KN_{\lambda}$ are
distinct, which implies the injectivity of $f$. By the compactness of
$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/K$, $f$ is closed, hence
$f^{-1}$ is continuous, thus a homeomorphism. ∎

######  Proposition 4.11.

(profinite Schurian association scheme) Let
${{{G}^{{\kern-1.04996pt}\wedge}}}=\varprojlim
G_{\lambda}=\varprojlim{{{G}^{{\kern-1.04996pt}\wedge}}}/N_{\lambda}$ be a
profinite group. We denote by $G_{\lambda}$ the quotient
${{{G}^{{\kern-1.04996pt}\wedge}}}/N_{\lambda}$. For a closed subgroup $H<G$,
its image in $G_{\lambda}$ is $H_{\lambda}:=HN_{\lambda}$. Then we have a
projective system of Schurian association schemes

$X_{\lambda}:=G_{\lambda}/H_{\lambda}$

and

$I_{\lambda}:=H_{\lambda}\backslash G_{\lambda}/H_{\lambda}.$

These systems give a profinite association scheme, which may be non-
commutative. We call this type of profinite association scheme Schurian. We
have canonical homeomorphisms

${{{G}^{{\kern-1.04996pt}\wedge}}}/H\stackrel{{\scriptstyle\sim}}{{\to}}{{{X}^{{\kern-1.04996pt}\wedge}}}=\varprojlim
X_{\lambda}=\varprojlim{{{G}^{{\kern-1.04996pt}\wedge}}}/HN_{\lambda}$

and

$H\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/H\stackrel{{\scriptstyle\sim}}{{\to}}{{{I}^{{\kern-1.04996pt}\wedge}}}=\varprojlim
I_{\lambda}=\varprojlim
H_{\lambda}\backslash{{{G}^{{\kern-1.04996pt}\wedge}}}/H_{\lambda}.$

###### Proof.

Everything is proved in Proposition 4.2 and Lemma 4.10, except that these
mappings give surjective morphisms of association schemes. Take
$\lambda\geq\mu$. Since the surjectivity is proved in (2), it suffices to show
the commutativity of

$\begin{matrix}X_{\lambda}\times X_{\lambda}&\to&I_{\lambda}\\\
\downarrow&&\downarrow\\\ X_{\mu}\times X_{\mu}&\to&I_{\mu},\end{matrix}$

which is easy to check: an element $(gHN_{\lambda},g^{\prime}HN_{\lambda})$ at
the left top is mapped to the right $Hg^{-1}g^{\prime}HN_{\lambda}$, then to
the bottom $Hg^{-1}g^{\prime}HN_{\mu}$. This is the same via the left bottom
corner. ∎

######  Proposition 4.12.

Let ${{{G}^{{\kern-1.04996pt}\wedge}}}/H$ be a Schurian profinite association
scheme as in 4.11. Suppose that each finite association scheme $X_{\lambda}$
is commutative. Then, ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is the set of
irreducible representations of ${{{G}^{{\kern-1.04996pt}\wedge}}}$ appearing
in $C(X_{\lambda})$ for some $\lambda$, where the multiplicity is at most one
for any $C(X_{\lambda})$.

###### Proof.

We have an inductive system of injections $C(X_{\lambda}\times X_{\lambda})$,
which gives via (4.2) an inductive system of injections

$\operatorname{Hom}_{\mathbb{C}}(C(X_{\mu}),C(X_{\mu}))\to\operatorname{Hom}_{\mathbb{C}}(C(X_{\lambda}),C(X_{\lambda}))$

for $\lambda\geq\mu$. To obtain $A_{X_{\mu}}$ and $A_{X_{\lambda}}$, we take
${{{G}^{{\kern-1.04996pt}\wedge}}}$-invariant part (4.3)

$\operatorname{Hom}_{G_{\mu}}(C(X_{\mu}),C(X_{\mu}))\to\operatorname{Hom}_{G_{\lambda}}(C(X_{\lambda}),C(X_{\lambda}))$

(note that there is a canonical surjection
${{{G}^{{\kern-1.04996pt}\wedge}}}\to G_{\lambda}$, hence a
$G_{\lambda}$-module is a ${{{G}^{{\kern-1.04996pt}\wedge}}}$-module.) By the
surjectivity of ${{{G}^{{\kern-1.04996pt}\wedge}}}\to G_{\lambda}$, an
irreducible representation in $C(X_{\mu})$ of $G_{\mu}$ lifts to an
irreducible representation in $C(X_{\lambda})$ of $G_{\lambda}$.

We assume that every finite association scheme appeared is commutative. Then,
the multiplicity of each irreducible representation $V\subset C(X_{\lambda})$
is one by Corollary 4.8 for any $\lambda$. Thus, the partial surjection given
in Corollary 2.14 is in this case the inverse to the injection $J_{\mu}\to
J_{\lambda}$, induced by $G_{\lambda}\to G_{\mu}$, and

${{{J}^{{\kern-1.04996pt}\wedge}}}=\varinjlim J_{\lambda}$

is identified with the set of irreducible representations of
${{{G}^{{\kern-1.04996pt}\wedge}}}$ appeared in some $C(X_{\lambda})$. ∎

Indeed, the above irreducible representations are exactly the irreducible
representations of ${{{G}^{{\kern-1.04996pt}\wedge}}}$ appearing in
$C({{{G}^{{\kern-1.04996pt}\wedge}}}/H)$. To show this, we recall a well known
“no small subgroups” lemma.

######  Lemma 4.13.

Let $G$ be a Lie group over ${\mathbb{R}}$ or ${\mathbb{C}}$. Then, there is
an open neighborhood of the unit $e$, which contains no subgroup except
$\\{e\\}$.

###### Proof.

We consider the Lie algebra $\mathfrak{g}$ of $G$. Give an arbitrary
(Hermitian or positive definite) metric to $\mathfrak{g}$. It is known that

$\exp:\mathfrak{g}\to G$

is homeomorphism when restricted to a small enough neighborhood of $0$. We
take the homeomorphic neighborhoods $U$ of $e\in G$ and $V$ of
$0\in\mathfrak{g}$. We may assume that $V$ is bounded with respect to the
metric. Assume that $U$ contains non-trivial subgroup of $G$. Take an $e\neq
g\in G$ in the subgroup. Consequently, for any integer $n$, $g^{n}\in U$. If
$h=\log(g)\in V\subset\mathfrak{g}$, $nh=\log(g^{n})$ corresponds to $g^{n}$
and hence $nh\in V$ for any $n$. However, $V$ is bounded, which contradicts
the assumption. ∎

######  Corollary 4.14.

Let ${{{G}^{{\kern-1.04996pt}\wedge}}}$ be a profinite group, and
$f:{{{G}^{{\kern-1.04996pt}\wedge}}}\to G$ a group morphism to a real or
complex Lie group $G$. Then, $\operatorname{Ker}(f)$ is an open normal
subgroup of ${{{G}^{{\kern-1.04996pt}\wedge}}}$, and the image
$f({{{G}^{{\kern-1.04996pt}\wedge}}})$ in $G$ is a finite subgroup of $G$.

###### Proof.

Take a small enough neighborhood of $U$ in Lemma 4.13. Consider
$H=f^{-1}(U)\subset{{{G}^{{\kern-1.04996pt}\wedge}}}$, which is open in
${{{G}^{{\kern-1.04996pt}\wedge}}}$. Thus, there exists $\lambda$ such that
the open normal subgroup $N=N_{\lambda}$ is contained in $H$. The image
$f(N)<G$ is a subgroup in $U$, which must be $\\{e\\}\subset G$ by definition
of $U$. Thus, $\operatorname{Ker}(f)$ contains $N$, and is a union of cosets
of $N$, hence an open normal subgroup of ${{{G}^{{\kern-1.04996pt}\wedge}}}$.
Since ${{{G}^{{\kern-1.04996pt}\wedge}}}$ is compact and $\operatorname{Ker}f$
is open,
$f({{{G}^{{\kern-1.04996pt}\wedge}}})\cong{{{G}^{{\kern-1.04996pt}\wedge}}}/\operatorname{Ker}f$
is a finite group. ∎

######  Proposition 4.15.

Let $G$ be a compact Hausdorff group, and $H$ a closed subgroup. Consider the
representation of $G$ on $C(G/H)$. Then, each irreducible component of
$C(G/H)$ is finite dimensional.

###### Proof.

This follows from a variant of the Peter-Weyl theorem, see Takeuchi[24,
Section 1]. ∎

######  Proposition 4.16.

Let ${{{G}^{{\kern-1.04996pt}\wedge}}}$ be a profinite group and $H$ a closed
subgroup. Consider the profinite association scheme arising from
${{{G}^{{\kern-1.04996pt}\wedge}}}/H$. Let
$J({{{G}^{{\kern-1.04996pt}\wedge}}}/H)$ be the set of the irreducible
representations of ${{{G}^{{\kern-1.04996pt}\wedge}}}$ on
$C({{{G}^{{\kern-1.04996pt}\wedge}}}/H)$. Then, all finite association schemes
$X_{\lambda}$ in Proposition 4.11 are commutative, if and only if the
representation $C({{{G}^{{\kern-1.04996pt}\wedge}}}/H)$ is multiplicity free.
In this case, there is a canonical bijection

$f:J({{{G}^{{\kern-1.04996pt}\wedge}}}/H)\stackrel{{\scriptstyle\sim}}{{\to}}{{{J}^{{\kern-1.04996pt}\wedge}}}=\varinjlim
J_{\lambda}.$ (4.5)

###### Proof.

Put ${{{X}^{{\kern-1.04996pt}\wedge}}}:={{{G}^{{\kern-1.04996pt}\wedge}}}/H$.
Let $V\subset C({{{X}^{{\kern-1.04996pt}\wedge}}})$ be an irreducible
representation of ${{{G}^{{\kern-1.04996pt}\wedge}}}$. By Proposition 4.15,
$V$ is finite dimensional. By Lemma 4.13, the representation factors through

$G\to G/N_{\lambda}\to{\mathrm{GL}}(V)$

for some $\lambda$. By Lemma 4.1, this means that $V\subset
C({{{X}^{{\kern-1.04996pt}\wedge}}})$ lies in

$V\subset
C(N_{\lambda}\backslash{{{X}^{{\kern-1.04996pt}\wedge}}})=C(X_{\lambda})\subset
C({{{X}^{{\kern-1.04996pt}\wedge}}}).$

Thus, $C({{{X}^{{\kern-1.04996pt}\wedge}}})$ is multiplicity free, if and only
if $C(X_{\lambda})$ is multiplicity free for every $\lambda\in\Lambda$, in
other words, if and only if every $X_{\lambda}$ is a commutative association
scheme, by Proposition 4.12. In the commutative case, the following
observation shows that every irreducible component of
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$ appears in $C(X_{\lambda})$ for some
$\lambda$. Because the multiplicity is one, for any $V$ above, there is a
unique representation isomorphic to $V$ in $\varinjlim C(X_{\lambda})$. This
gives a map $f$ in (4.5). The injectivity follows from that
$C({{{X}^{{\kern-1.04996pt}\wedge}}})$ is multiplicity-free. The surjectivity
follows from $C(X_{\lambda})\subset C({{{X}^{{\kern-1.04996pt}\wedge}}})$ is a
${{{G}^{{\kern-1.04996pt}\wedge}}}$-submodule, and any
$G_{\lambda}$-irreducible component of $C(X_{\lambda})$ is a
${{{G}^{{\kern-1.04996pt}\wedge}}}$-irreducible component since
${{{G}^{{\kern-1.04996pt}\wedge}}}\to G_{\lambda}$ is surjective. ∎

######  Remark 4.17.

Kurihara-Okuda[13] gives the notion of Bose-Mesner algebra for general
homogeneous space $G/H$, where $G$ is compact Hausdorff and $H$ is a closed
subgroup. In the case where $G$ is a profinite group and $C(G/H)$ is
multiplicity free, their construction coincides with our definition, by
Proposition 4.16.

####  4.1.2. Profinite abelian groups

For any finite abelian group $X$, we consider its thin scheme. Namely,
$R:X\times X\to I=X$ with $(x,y)\mapsto y-x$. Then, the $i$-th adjacency
matrix $R^{-1}(i)$ is the representation matrix of addition of $i$ on $C(X)$,
and the Bose-Mesner algebra is isomorphic to the group ring $C(X)$, with usual
Hadamard product and the convolution product

$(f*g)(x)=\frac{1}{\\#X}\sum_{y}f(x-y)g(y)$

(the factor $\frac{1}{\\#X}$ is not the standard, incorporated to be
compatible with Theorem 2.5). Primitive idempotents are exactly the characters
$\xi\in{{\check{X}}}$, where ${{\check{X}}}\subset C(X)$ is the group of
characters of $X$.

For a profinite abelian group ${{{X}^{{\kern-1.04996pt}\wedge}}}$, it is a
projective limit of finite abelian groups $X_{\lambda}$. We have an inductive
system of the character groups $J_{\lambda}:=\check{X_{\lambda}}$. For
$\lambda\geq\mu$, the partial map $p_{\lambda,\mu}:J_{\lambda}\rightharpoonup
J_{\mu}$ is given by the injection $i_{\mu,\lambda}:J_{\mu}\to J_{\lambda}$,
where the domain of $p_{\lambda,\mu}$ is the image of $i_{\mu,\lambda}$, and
in the domain $p_{\lambda,\mu}$ is the inverse of $i_{\mu,\lambda}$. Thus,
${{{I}^{{\kern-1.04996pt}\wedge}}}$ is isomorphic to
${{{X}^{{\kern-1.04996pt}\wedge}}}$, ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is
the inductive limit of the character groups $\check{X_{\lambda}}$. The Bose-
Mesner algebra is isomorphic to $C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$,
the space of finite linear combinations of characters of
${{{X}^{{\kern-1.04996pt}\wedge}}}$, which is isomorphic to
$C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$, the space of locally constant
functions on
${{{I}^{{\kern-1.04996pt}\wedge}}}\simeq{{{X}^{{\kern-1.04996pt}\wedge}}}$.
This example is outside of the scope of Barg-Skriganov theory, if
${{{I}^{{\kern-1.04996pt}\wedge}}}\simeq{{{X}^{{\kern-1.04996pt}\wedge}}}$ is
uncountable.

######  Remark 4.18.

Since ${{{J}^{{\kern-1.04996pt}\wedge}}}$ is discrete,
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$ has a natural (and orthogonal)
basis consisting of $E_{j}$ ($j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$), but for
the canonically isomorphic vector space
$C_{lc({{{I}^{{\kern-0.74997pt}\wedge}}})}$, we do not have a natural basis,
since ${{{I}^{{\kern-1.04996pt}\wedge}}}$ is compact with (possibly)
uncountable cardinality. This makes a definition of eigenmatrices difficult.

###  4.2. Kernel schemes

Another example of profinite association schemes is a projective system of the
kernel schemes. The kernel schemes are finite association schemes defined in
Martin-Stinson[16] to study $(t,m,s)$-nets. The $(t,m,s)$-nets are introduced
by Niederreiter for quasi-Monte Carlo integrations, see [18].

We recall the kernel scheme, with a slight modification on the notation for
$I$, to make a description as a projective system easier.

######  Definition 4.19.

Let $n$ be a positive integer, and $V$ be a finite set of alphabet with
cardinality $v\geq 2$. Let $X_{n}$ be $V^{n}$, and
$I_{n}:=\\{1,2,\ldots,n\\}\cup\\{\infty\\}$. Define $R_{n}:X_{n}\times
X_{n}\to I_{n}$ as follows. Let $x=(x_{1},x_{2},\ldots,x_{n})$ and
$y=(y_{1},y_{2},\ldots,y_{n})$ be elements of $X_{n}$. Let $R(x,y)$ be the
smallest index $i$ for which $x_{i}\neq y_{i}$. If $x=y$, then
$R(x,y)=\infty$. This is a symmetric (and hence commutative) association
scheme, with $R^{-1}(\infty)$ being the identity relation. This is called a
kernel scheme, and denoted by $\overrightarrow{k(n,v)}$.

A direct computation of the intersection numbers shows that this is a
commutative association scheme, as shown in [16]. We shall give an indirect
proof in Corollary 4.27. (This is only for the self-containedness.)

######  Proposition 4.20.

The kernel schemes $\overrightarrow{k(n,v)}$ forms a projective system, where
$X_{n+1}\to X_{n}$ is given by deleting the right most component, and
$I_{n+1}\to I_{n}$ is given by $m\mapsto m$ for $m\leq n$, and the both
$n+1,\infty$ are mapped to $\infty$. Then
${{{X}^{{\kern-1.04996pt}\wedge}}}=V^{{\mathbb{N}}_{>0}}$ holds, and
${\operatorname{pr}}_{n}:{{{X}^{{\kern-1.04996pt}\wedge}}}\to X_{n}$ is
obtained by taking the left (first) $n$ components, while
${{{I}^{{\kern-1.04996pt}\wedge}}}$ is ${\mathbb{N}}_{>0}\cup\\{\infty\\}$,
with topology obtained by: each $n\in{\mathbb{N}}_{>0}$ is clopen, and an open
set containing $\infty$ is a complement of a finite subset of
${\mathbb{N}}_{>0}$, that is, ${{{I}^{{\kern-1.04996pt}\wedge}}}$ is the one-
point compactification of ${\mathbb{N}}_{>0}$. We call this a pro-kernel
scheme and denote by $\overrightarrow{k(\infty,v)}$.

###### Proof.

It is easy to check that these make a projective system of association
schemes. The topology of ${{{I}^{{\kern-1.04996pt}\wedge}}}$ comes from the
definition of projective limit topology. ∎

######  Remark 4.21.

In this case ${{{I}^{{\kern-1.04996pt}\wedge}}}$ is countable, and this is a
special case of infinite association schemes examined in detail, named metric
schemes, by Barg-Skriganov [2, Section 8], for which the adjacency algebra is
defined. In particular, in (8.33) they showed that it is an inductive limit of
finite dimensional adjacency algebra, which indeed coincides with our
definition.

######  Remark 4.22.

This is merely an observation. For the set of $p$-adic integers
${\mathbb{Z}}_{p}$, we have a valuation
$v:{\mathbb{Z}}_{p}\to{\mathbb{N}}\cup\\{\infty\\}$ giving an ultrametric. The
above pro-kernel scheme is isomorphic to this metric, up to the difference by
one on the metric, i.e.,

$R:{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}\to{\mathbb{N}}\cup\\{\infty\\},(x,y)\mapsto
v(x-y).$

The same holds for a ring of formal power series ${\mathbb{F}}_{q}[[t]]$,
where ${\mathbb{F}}_{q}$ denotes the finite field of $q$ elements. The both
yield the isomorphic pro-kernel schemes if $p=q$.

To see ${{{J}^{{\kern-1.04996pt}\wedge}}}$ of the pro-kernel scheme, we need
to work with primitive idempotents. All necessary results are in Martin-
Stinson[16], but because of the choice of the index $I$ and normalization of
the products necessary for making the projective system, the obtained
constants slightly differ. To help the readers’ understandings, we recall the
methods of Martin-Stinson. For the chosen integer $v\geq 2$, let $V:=Z_{v}$ be
the additive group ${\mathbb{Z}}/v$. Then $X=Z_{v}^{n}$ is an additive group,
and the kernel scheme is a translation scheme [4, §2.10].

######  Definition 4.23.

Let $X$ be a finite abelian group. A translation scheme is an association
scheme $R:X\times X\to I$, which factors through $X\times X\to X,(x,y)\mapsto
y-x$, namely, there is an $S:X\to I$ such that $R(x,y)=S(y-x)$.

######  Definition 4.24.

Return to the kernel scheme. We define ${{\operatorname{bot}}}:X_{n}\to I_{n}$
by

${{\operatorname{bot}}}(x_{1},\ldots,x_{n}):=\min\\{i\mid x_{i}\neq 0\\},$

and ${{\operatorname{bot}}}(0,\cdots,0)=\infty$. Then we have

$R_{n}:X_{n}\times X_{n}\to
I_{n},\quad(x,y)\mapsto{{\operatorname{bot}}}(x-y).$

By $S:={{\operatorname{bot}}}$, the kernel scheme is a translation scheme.

Let $(X,R,I)$ be a translation scheme. For $i\in I$, let $S_{i}\subset X$ be
$S^{-1}(i)$ ($S$ for the sphere). Let ${{\check{X}}}$ be the dual group of
$X$. Then, $\xi\in{{\check{X}}}$ is an orthonomal basis of $C(X)$ under the
normalized inner product in Definition 3.16. Let $A_{i}$ denote the $i$-th
adjacency matrix, and $\chi_{i}:X\to\\{0,1\\}$ be the indicator function for
$S_{i}$. Our action in Theorem 2.5 shows that for $x\in X$

$\displaystyle(A_{i}\bullet\xi)(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in X}A_{i}(x,y)\xi(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in X}\chi_{i}(y-x)\xi(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{y\in x+S_{i}}\xi(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}\sum_{a\in S_{i}}\xi(x+a)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#X}(\sum_{a\in S_{i}}\xi(a))\xi(x).$

Thus $\xi$ is an eigenvector with eigenvalue

$\frac{1}{\\#X}(\sum_{a\in S_{i}}\xi(a)).$ (4.6)

The primitive idempotents are projectors to the common eigen spaces of $A_{i}$
for all $i\in I$. Let us assume $X=Z_{v}^{n}$, and then
${{\check{X}}}={{\check{Z_{v}}}}^{n}$, where the product of
${{\check{Z_{v}}}}$ is written multiplicatively with unit $1$. Thus,

$(\xi_{1},\ldots,\xi_{n})(x_{1},\ldots,x_{n})=\prod_{i=1}^{n}\xi_{i}(x_{i}).$
(4.7)

######  Definition 4.25.

For $\xi=(\xi_{1},\ldots,\xi_{n})\in{{\check{X}}}$, define
${{\operatorname{top}}}(\xi)$ as the max $i$ such that $\xi_{i}\neq 1$. If
$\xi=1$, then ${{\operatorname{top}}}(\xi)=0$. Thus
${{\operatorname{top}}}:{{\check{X}}}\to\\{0,1,\ldots,n\\}$. We shall denote
$J_{n}:=\\{0,1,\ldots,n\\}$, because we shall soon show a natural
correspondence between the set of primitive idempotents and $J_{n}$.

######  Lemma 4.26.

For $\xi$ with $j={{\operatorname{top}}}(\xi)$, we denote by $p_{i}(j)$ the
eigenvalue of $A_{i}$ for $\xi$. Then

$p_{i}(j)=\begin{cases}v^{-n}&\mbox{ if }i=\infty,\\\ (v-1)v^{-i}&\mbox{ if
}j<i<\infty,\\\ -v^{-i}&\mbox{ if }j=i,\\\ 0&\mbox{ if }j>i.\\\ \end{cases}$

Thus, for any $A_{i}$, the eigenvalue of $\xi\in{{\check{X}}}$ depends only on
${{\operatorname{top}}}(\xi)$, and if
${{\operatorname{top}}}(\xi)\neq{{\operatorname{top}}}(\xi^{\prime})$, then
there is an $i$ such that $A_{i}$ has different eigenvalues for $\xi$ and
$\xi^{\prime}$.

###### Proof.

Recall that $I=\\{1,2,\ldots,n\\}\cup\\{\infty\\}$. Put

$B_{i}:=\bigcup_{i\leq\ell\leq\infty}S_{\ell}.$

($B_{i}$ is the ball consisting of all the elements $(x_{1},\ldots,x_{n})$
with $x_{\ell}=0$ for all $\ell<i$, with the cardinality $v^{n-i+1}$ for
$i<\infty$ and $1$ for $i=\infty$.) Fix $i$ and $\xi\in{{\check{X}}}$ with
${{\operatorname{top}}}(\xi)=j$, then by (4.6) we have

$\displaystyle\tau(i,j)\xi$ $\displaystyle:=$
$\displaystyle\sum_{i\leq\ell\leq\infty}A_{i}\bullet\xi$ $\displaystyle=$
$\displaystyle\frac{1}{v^{n}}(\sum_{a\in B_{i}}\xi(a))\xi$ $\displaystyle=$
$\displaystyle\begin{cases}v^{-n}\xi&\mbox{ if }i=\infty,\\\
v^{-i+1}\xi&\mbox{ if }j<i<\infty,\\\ 0&\mbox{ if }j\geq i.\\\ \end{cases}$

The last equality follows from that for $i=\infty$, $B_{\infty}=\\{0\\}$ and
hence the summation is $\sum_{a\in B_{i}}\xi(a)=1$, and if $j<i<\infty$, then
(4.7) is always $1$ and the cardinality of $B_{i}$ is $v^{n-i+1}$, while if
$j\geq i$, $\xi(a)$ depends on $a$ and the summation over $a\in B_{i}$ is
zero, since $\xi$ is a non-trivial character on the abelian group $B_{i}$.
Thus, the eigenvalue of $A_{i}$ for $\xi$ is the summation over
$S_{i}=B_{i}\setminus B_{i+1}$, that is,

$\begin{cases}v^{-n}&\mbox{ if }i=\infty,\\\
(\tau(i,j)-\tau(i+1,j))=(v-1)v^{-i}&\mbox{ if }j<i<\infty,\\\
(\tau(i,j)-\tau(i+1,j))=-v^{-i}&\mbox{ if }i=j,\\\ 0&\mbox{ if
}j>i,\end{cases}$

as desired. ∎

######  Corollary 4.27.

The kernel scheme $\overrightarrow{k(n,v)}$ is a commutative association
scheme. A primitive idempotent is the projection to the subspace $C(X)_{j}$ of
$C(X)$ spanned by $\\{\xi\mid{{\operatorname{top}}}(\xi)=j\\}$ for each $j\in
J_{n}$, which identifies the set of primitive idempotents with $J_{n}$.

###### Proof.

Recall that we did not use the property of association schemes for the kernel
schemes so far. Every $A_{i}$ has $C(X)_{j}$ as an eigenspace with an
eigenvalue. Since $C(X)$ is a direct sum of $C(X)_{j}$ for $j\in J_{n}$, this
implies that the matrices $A_{i}$ mutually commute. We consider the algebra
generated by $A_{i}$ by the matrix product. Each element of this algebra is
determined by the $n+1$ eigenvalues corresponding to $J_{n}$. Thus, the
dimension of the algebra as a vector space does not exceed $n+1$, but
$A_{i}$’s are $n+1$ linearly independent matrices, hence their linear
combination is closed under the matrix multiplication. The rest of the axioms
of association schemes are easy to verify. The description of primitive
idempotents immediately follows. ∎

######  Corollary 4.28.

The multiplicity $m_{j}$ is $1$ for $j=0$ and $m_{j}=(v-1)v^{j-1}$ for $j>0$.
The valency $k_{i}$ is $1$ for $i=\infty$ and $k_{i}=(v-1)v^{n-i}$ for
$i<\infty$.

###### Proof.

The multiplicity is the number of $\xi$ with ${{\operatorname{top}}}(\xi)=j$,
which implies the first statement. The valency is the cardinality of the
sphere $S_{i}$, which implies the second. ∎

######  Corollary 4.29.

For $j\in J_{n}$, the projector $E_{j}$ is given by

$\displaystyle E_{j}(x,y)$ $\displaystyle=$
$\displaystyle\sum_{\xi\in{{\check{X}}},{{\operatorname{top}}}(\xi)=j}\xi(x)\overline{\xi(y)}$
$\displaystyle=$
$\displaystyle\sum_{\xi\in{{\check{X}}},{{\operatorname{top}}}(\xi)=j}\xi(x-y).$

We have

$A_{i}=\sum_{j\in J_{n}}p_{i}(j)E_{j}$

where $p_{i}(j)$ is defined in Lemma 4.26.

###### Proof.

The first half describes the projection to the eigenspace corresponding to $j$
by an orthonomal basis. The second is because $A_{i}$ has the eigenvalue
$p_{i}(j)$ for that eigenspace. ∎

For $x-y\in S_{i}$, we define

$\displaystyle q_{j}(i)$ $\displaystyle:=$ $\displaystyle
E_{j}(x,y)=\sum_{\xi\in{{\check{X}}},{{\operatorname{top}}}(\xi)=j}\xi(x-y)$
$\displaystyle=$ $\displaystyle\begin{cases}1&\mbox{ if }0=j\\\
(v-1)v^{j-1}&\mbox{ if }0<j<i\leq\infty\\\ -v^{j-1}&\mbox{ if }j=i\\\ 0&\mbox{
if }j>i.\end{cases}$

This computation is dual to that for $p_{i}(j)$, so omitted.

######  Corollary 4.30.

$E_{j}=\sum_{i\in I}q_{j}(i)A_{i}.$

###### Proof.

We evaluate the both sides at $(x,y)$. The left hand side is $q_{j}(i)$ for
$x-y\in S_{i}$, which is equal to the right hand side, since
$A_{i}(x,y)=\chi_{i}(x-y)$. ∎

######  Corollary 4.31.

In $\overrightarrow{k(\infty,v)}$, the set ${{{J}^{{\kern-1.04996pt}\wedge}}}$
is the inductive limit of $J_{n}\to J_{n+1}$, and hence naturally identified
with ${\mathbb{N}}$.

Now we pass to the pro-kernel scheme, i.e., take the inductive limit of
$A_{\overrightarrow{k(n,v)}}$ to obtain
$A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}=A_{\overrightarrow{k(\infty,v)}}$.

######  Proposition 4.32.

Let $X_{n}$ denote the kernel scheme $\overrightarrow{k(n,v)}$ with $I_{n}$
and primitive idempotents $J_{n}$, and ${{{X}^{{\kern-1.04996pt}\wedge}}}$ the
pro-kernel scheme $\overrightarrow{k(\infty,v)}$ with
${{{I}^{{\kern-1.04996pt}\wedge}}}$ and the primitive idempotents
${{{J}^{{\kern-1.04996pt}\wedge}}}$. For
$i\in{{{I}^{{\kern-1.04996pt}\wedge}}}\setminus\\{\infty\\}$, we denote by
$A_{i}$ the image of $\delta_{i}\in C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$
by the isomorphism $C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})\to
A_{{{X}^{{\kern-0.74997pt}\wedge}}}$. Note that $\delta_{\infty}\notin
C_{lc({{{I}^{{\kern-0.74997pt}\wedge}}})}$. Through $A_{X_{n}}\to
A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$, $E_{j}$ is mapped to $E_{j}$, and
$A_{i}$ for $i\in\\{1,2,\ldots,n\\}$ is mapped to $A_{i}$, while
$A_{\infty}\in A_{X_{n}}$ is mapped to an element corresponding to the
indicator function $\chi_{\\{n+1,\ldots,\infty\\}}\in
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$. It is convenient to denote the
corresponding element in $A_{{{X}^{{\kern-0.74997pt}\wedge}}}$ by the symbol

$\sum_{i\in\\{n+1,\ldots,\infty\\}}A_{i},$

since

$\chi_{\\{n+1,\ldots,\infty\\}}=\sum_{i\in\\{n+1,\ldots,\infty\\}}\delta_{i}$

holds. (Note that $A_{\infty}$ does not exist in
$A_{{{X}^{{\kern-0.74997pt}\wedge}}}$.)

###### Proof.

According to Proposition 4.20 we have
${{{I}^{{\kern-1.04996pt}\wedge}}}={\mathbb{N}}_{>0}\cup\\{\infty\\}$ and
${{{J}^{{\kern-1.04996pt}\wedge}}}={\mathbb{N}}$ by Corollary 4.31. Since
$J_{n}\to{{{J}^{{\kern-1.04996pt}\wedge}}}$ is an injection, $E_{j}\in
A_{X_{n}}$ is mapped to $E_{j}\in A_{{{X}^{{\kern-0.74997pt}\wedge}}}$ (see
Proposition 3.24). The projection ${{{I}^{{\kern-1.04996pt}\wedge}}}\to I_{n}$
is obtained by mapping $i\mapsto i$ for $1\leq i\leq n$, and $i\mapsto\infty$
for $i>n$. Since the indicator function $\delta_{i}\in C(I_{n})$ is mapped to
$\delta_{i}\in C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$ for $1\leq i\leq n$,
$A_{i}$ is mapped to $A_{i}$. The description of the image of $A_{\infty}$ is
by the definition of $A_{X_{n}}\to A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$,
which comes from ${{{I}^{{\kern-1.04996pt}\wedge}}}\to I_{n}$. ∎

Everything is well-defined, except
$i_{0}=\infty\in{{{I}^{{\kern-1.04996pt}\wedge}}}$. In fact, the operators
$A_{i}$, $E_{j}$, the values $p_{i}(j)$, $q_{j}(i)$,
$\mu_{{{J}^{{\kern-0.74997pt}\wedge}}}(j)=m_{j}$,
$\mu_{{{I}^{{\kern-0.74997pt}\wedge}}}(i)=k_{i}/\\#X=(v-1)v^{-i}$ are all
well-defined, except that $A_{\infty}$ and $p_{\infty}(j)$ are not well-
defined, since there is no corresponding element to $A_{\infty}$ in
$A_{{{X}^{{\kern-0.74997pt}\wedge}}}$. Indeed, $A_{\infty}$ would correspond
to the indicator function of $\infty\in{{{I}^{{\kern-1.04996pt}\wedge}}}$,
which is not continuous and hence is outside
$A_{{{X}^{{\kern-0.74997pt}\wedge}}}$. This forces to choose something which
replaces $A_{\infty}$. One possible way is to replace it with $E_{0}$ because

$E_{0}=\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}A_{i}$

holds as a function (constant value $1$) on
${{{I}^{{\kern-1.04996pt}\wedge}}}$. then

$\\{A_{i}\mid i\in{\mathbb{N}}_{>0}\\}\cup\\{E_{0}\\}$

is a linear basis of $C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$, although
this is not an orthogonal basis. This is the method chosen in [2, (8.3),
(8.9)] to replace an infinite sum with a finite sum ($\alpha_{0}$ in (8.8) in
their paper is nothing but $E_{0}$), which gives a precise definition of the
adjacency algebra. In our interpretation in Proposition 4.32, the expression
in Corollary 4.29 becomes

$A_{i}=\sum_{j\in{{{J}^{{\kern-0.74997pt}\wedge}}}}p_{i}(j)E_{j}$

which is a finite sum (see Lemma 4.26) and holds for $i\neq\infty$ (since the
equality holds in $A_{X_{i}}$), except the problem that for $i=\infty$, the
value $p_{\infty}(j)=v^{-n}$ converges to $0$, while the summation is over
infinitely many $E_{j}$. In Corollary 4.30 the expression

$E_{j}=\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}q_{j}(i)A_{i}$

seems an infinite sum, but $q_{j}(i)=(v-1)v^{j-1}$ is constant for $i>j$, and
thus the right hand side lies in $C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$,
or more precisely, the seemingly infinite sum is a finite sum by the usage of
the symbol in Proposition 4.32, and the equality does hold (since it holds in
$A_{X_{j}}$.) This may be translated into a finite summation formula for
$j\geq 1$:

$\displaystyle E_{j}$ $\displaystyle=$ $\displaystyle\sum_{i\leq
j}(q_{j}(i)-(v-1)v^{j-1})A_{i}+(v-1)v^{j-1}E_{0}$ $\displaystyle=$
$\displaystyle(\sum_{i<j}-(v-1)v^{j-1}A_{i})-v^{j}A_{j}+(v-1)v^{j-1}E_{0}.$

###  4.3. Ordered Hamming schemes

Martin and Stinson [16, Section 1.3] introduced ordered Hamming schemes as
Delsarte’s extension of length $s$ [6, Section 2.5] of the kernel schemes.

######  Proposition 4.33.

(Extension of length $s$)

Let $R:X\times X\to I$ be a commutative association scheme. Let $s$ be a
positive integer, and $S_{s}$ a symmetric group of degree $s$, acting on the
direct product $I^{s}$. We obtain a quotient map $I^{s}\to I^{s}/S_{s}$. Then,
the composition

$X^{s}\times X^{s}\to I^{s}\to I^{s}/S_{s}$

is a commutative association scheme. For $\bar{i}\in I^{s}/S_{s}$, the
corresponding Hadamard primitive idempotent is the sum of

$A_{i_{1}}\otimes\cdots\otimes A_{i_{s}}$

over $(i_{1},\ldots,i_{s})\in I^{s}$ which maps to $\bar{i}$. The set of
primitive idempotents are naturally identified with $J^{s}/S_{s}$, as follows.
For $\bar{j}\in J^{s}/S_{s}$, the corresponding primitive idempotent is the
sum of

$E_{j_{1}}\otimes\cdots\otimes E_{j_{s}}$

over $(j_{1},\ldots,j_{s})\in J^{s}$ which maps to $\bar{j}$. The
corresponding eigenspace is the direct sum of the tensors of the eigenspaces

$C(X)_{j_{1}}\otimes\cdots\otimes C(X)_{j_{s}}$

over $(j_{1},\ldots,j_{s})\in J^{s}$ which maps to $\bar{j}$.

A proof is not given in the paper by Delsarte, but found in an unpublished
(but reachable) paper by Godsil [9, 3.2 Corollary]. Because he did not mention
the eigenspaces, which we need later, we shall recall a proof.

###### Proof.

Let

$X^{s}\times X^{s}\to I^{s}$

be the direct product association scheme. Its Bose-Mesner algebra
$A_{X^{s}}=A_{X}^{\otimes s}$ is generated by the $s$-fold kronecker products
of $A_{i}$’s for $i\in I$, which satisfies the axiom of association scheme.
The symmetric group $S_{s}$ acts on $A_{X^{s}}$ by permutation. We take the
fixed part $A_{X^{s}}^{S_{s}}$ of $A_{X^{s}}$. Since $S_{s}$ preserves the two
products and the transpose, fix the $i_{0}$ and the $j_{0}$,
$A_{X^{s}}^{S_{s}}$ is a Bose-Mesner algebra. As a vector space, this is
canonically isomorphic to $C(I^{s})^{S_{s}}\cong C(I^{s}/S_{s})$, induced by
$I^{s}\to I^{s}/S_{s}$. For an $\bar{i}\in I^{s}/S_{s}$, its inverse image is
the set of $(i_{1},\ldots,i_{s})$ mapped to $\bar{i}$, and the set of
$A_{i_{1}}\otimes\cdots\otimes A_{i_{s}}$ are mutually disjoint Hadamard
idempotent, whose sum is an Hadamard idempotent $A_{\bar{i}}$. This must be
primitive in $A_{X^{s}}^{S_{s}}$, since the Hadamard product satisfies
$A_{\bar{i}}\circ A_{\bar{i}^{\prime}}=\delta_{\bar{i}\ \bar{i}^{\prime}}$,
and their sum is the Hadamard unit $J_{A_{X^{s}}}$. The same argument applies
for the primitive idempotents $E_{\bar{j}}$. The description of the
corresponding eigenspace follows. ∎

This construction, when applied to the kernel schemes $X_{n}$,
$I_{n}=\\{1,2,\ldots,n\\}\cup\\{\infty\\}$, and $R_{n}:X_{n}\times X_{n}\to
I_{n}$ given Definition 4.19), yields an ordered Hamming scheme.

######  Definition 4.34.

The ordered Hamming scheme, denoted by $\overrightarrow{H}(s,n,v)$, is defined
as the extension of length $s$ of the kernel scheme $\overrightarrow{k(n,v)}$.

The following is easy to check.

######  Proposition 4.35.

The projections $X_{n+1}^{s}\to X_{n}^{s}$ and $I_{n+1}^{s}/S_{s}\to
I_{n}^{s}/S_{s}$ coming from Proposition 4.20 give a projective system
$\overrightarrow{H}(s,n+1,v)\to\overrightarrow{H}(s,n,v)$ of commutative
association schemes, namely, a profinite association scheme, which we call the
pro-ordered Hamming scheme and denote by $\overrightarrow{H}(s,\infty,v)$. The
${{{I}^{{\kern-1.04996pt}\wedge}}}$ of this projective association schemes is
$({\mathbb{N}}_{>0}\cup\\{\infty\\})^{s}/S_{s}$, and
${{{J}^{{\kern-1.04996pt}\wedge}}}$ is ${\mathbb{N}}^{s}/S_{s}$.

We shall mention on this profinite scheme later in the last of Section 5.3.

##  5\. Delsarte theory for profinite association schemes

Here we extend Delsarte theory introduced in [6]. The method here follows
Kurihara-Okuda [13], which generalizes Delsarte theory to compact homogeneous
spaces. In this section, let $(X_{\lambda},R_{\lambda},I_{\lambda})$ be a
profinite association scheme, and ${{{X}^{{\kern-1.04996pt}\wedge}}}$,
${{{I}^{{\kern-1.04996pt}\wedge}}}$, ${{{J}^{{\kern-1.04996pt}\wedge}}}$,
$A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$ be those defined in Section 3.1.

###  5.1. Multiset and Averaging functional

We consider a finite multi-subset $Y$ of ${{{X}^{{\kern-1.04996pt}\wedge}}}$,
which means that $Y$ is a set in which finite multiplicity of elements is
allowed and taken into account. To make the notion rigorous, we consider a map
from a finite set $Z$ to a set $X$

$g:Z\to X.$

Then the image $Y:=g(Z)$ in $X$ has a natural finite multiset structure, where
the multiplicity of $y\in Y$ is the cardinality of the fiber $\\#g^{-1}(y)$.
For $S\subset X$, we use the notation

$\\#(Y\cap S):=\\#(Z\cap g^{-1}(S)).$

This merely means to count the number of elements in $Y\cap S$ with taking the
multiplicity into account. Thus, we call $Y$ a “multi-subset” of $X$, and use
the notation

a finite multiset $Y\subset X$

by an abuse of language.

For any non-empty finite multiset $Y\subset{{{X}^{{\kern-1.04996pt}\wedge}}}$,
we would like to describe the notions of codes and designs. We begin with
defining the averaging functional.

######  Definition 5.1.

Let $Y\subset{{{X}^{{\kern-1.04996pt}\wedge}}}$ be a finite multi-subset in
the sense above. Define the averaging functional

$\operatorname{avg}_{Y}:C({{{X}^{{\kern-1.04996pt}\wedge}}})\to{\mathbb{C}},\quad
f\mapsto\frac{1}{\\#Y}\sum_{x\in Y}f(x):=\frac{1}{\\#Z}\sum_{z\in Z}f(g(z)).$

By (3.1), $C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})\cong
A_{{{X}^{{\kern-0.74997pt}\wedge}}}\cong
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$ holds, and by

$A_{{{X}^{{\kern-0.74997pt}\wedge}}}\to
C({{{X}^{{\kern-1.04996pt}\wedge}}}\times{{{X}^{{\kern-1.04996pt}\wedge}}})\stackrel{{\scriptstyle\operatorname{avg}_{Y}\otimes\operatorname{avg}_{Y}}}{{\to}}{\mathbb{C}},$

$\operatorname{avg}_{Y}^{2}:=\operatorname{avg}_{Y}\otimes\operatorname{avg}_{Y}$
defines a functional on $A_{{{X}^{{\kern-0.74997pt}\wedge}}}$, on
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$, and on
$C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$. For example, for $f\in
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$, we have

$\displaystyle\operatorname{avg}_{Y}^{2}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#Y^{2}}\sum_{x,y\in
Y}f\circ{{{R}^{{\kern-1.04996pt}\wedge}}}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#Y^{2}}\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}\sum_{x,y\in
Y,{{{R}^{{\kern-0.74997pt}\wedge}}}(x,y)=i}f(i)$ $\displaystyle=$
$\displaystyle\frac{1}{\\#Y^{2}}\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}\\#((Y\times
Y)\cap{{{R}^{{\kern-1.04996pt}\wedge}}}^{-1}(i))f(i).$

Note that these coefficients are the inner-distribution of $Y$ (Delsarte [6,
Section 3.1]) multiplied by $\frac{1}{\\#Y}$. Let us denote by
${\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}$ the vector space
whose basis is ${{{I}^{{\kern-1.04996pt}\wedge}}}$, namely, the space of
finite linear combinations of the elements of
${{{I}^{{\kern-1.04996pt}\wedge}}}$. We have a mapping

${\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}\to
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})^{\vee},$ (5.1)

where $\vee$ denotes the dual (i.e.
$\operatorname{Hom}(C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}}),{\mathbb{C}})$),
by the evaluation at $i$: $i\mapsto(f\mapsto f(i))$, which is injective since
only a finite number of linear combinations appear in
${\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}$, and their support
can be separated by clopen subsets. The above computation shows that
$\operatorname{avg}_{Y}^{2}\in
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})^{\vee}$ lies in
${\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}$, namely,

$\operatorname{avg}_{Y}^{2}=\sum_{i\in I}\frac{1}{\\#Y^{2}}(\\#((Y\times
Y)\cap{{{R}^{{\kern-1.04996pt}\wedge}}}^{-1}(i)))\cdot
i\in{\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}.$ (5.2)

Note that every coefficient is non-negative. Next we compute
$\operatorname{avg}_{Y}^{2}$ on $C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$.
For $j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$, we have the orthogonal projector
$E_{j}:C({{{X}^{{\kern-1.04996pt}\wedge}}})\to
C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$. For $f\in
C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$,

$f(x)=\int_{y\in{{{X}^{{\kern-0.74997pt}\wedge}}}}E_{j}(x,y)f(y)d\mu(y)=(f,\overline{E_{j}(x,-)})_{{\operatorname{HS}}}.$

Thus, we define $\operatorname{avg}_{Y}^{j}\in
C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$ by

$\operatorname{avg}_{Y}^{j}(-):=\frac{1}{\\#Y}\sum_{x\in Y}E_{j}(x,-),$ (5.3)

which represents the averaging functional in
$C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$:

$(f,\overline{\operatorname{avg}_{Y}^{j}})_{{\operatorname{HS}}}=\operatorname{avg}_{Y}(f).$
(5.4)

Then

$\displaystyle\operatorname{avg}_{Y}^{2}(E_{j})$ $\displaystyle=$
$\displaystyle\frac{1}{\\#Y^{2}}\sum_{x,y\in Y}E_{j}(x,y)$ (5.5)
$\displaystyle=$ $\displaystyle\frac{1}{\\#Y}\sum_{y\in
Y}\operatorname{avg}_{Y}^{j}(y)$ $\displaystyle=$
$\displaystyle(\operatorname{avg}_{Y}^{j},\overline{\operatorname{avg}_{Y}^{j}})=||\operatorname{avg}_{Y}^{j}||_{{\operatorname{HS}}}^{2}\geq
0.$

These positivities make the LP method by Delsarte possible [6, Thorem 3.3].

######  Definition 5.2.

Let us denote by $Q_{j}\in C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})$ the
image of $E_{j}\in A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$. This may be
considered as a description of the canonical isomorphism

$Q:C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})\to
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}}),\quad\delta_{j}\mapsto Q_{j},$

where $\delta_{j}$ means the indicator function on
${{{J}^{{\kern-1.04996pt}\wedge}}}$ at
$j\in{{{J}^{{\kern-1.04996pt}\wedge}}}$. (Note that $E_{j}\in
A_{{{{X}^{{\kern-0.74997pt}\wedge}}}}$ corresponds to $\delta_{j}\in
C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})$ in Theorem 3.13, see Corollary
3.11.) Then we have an injection

${\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}\to
C_{lc}({{{I}^{{\kern-1.04996pt}\wedge}}})^{\vee}\stackrel{{\scriptstyle
Q^{\vee}}}{{\to}}C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})^{\vee},$

obtained by

$i\mapsto(f\mapsto f(i))\mapsto(\delta_{j}\mapsto Q_{j}(i)).$

######  Definition 5.3.

The above morphism
$Q^{\vee}|_{{{I}^{{\kern-0.74997pt}\wedge}}}:{\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}\to
C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})^{\vee}$ is called the Mac-Williams
transform. In concrete, it maps

$Q^{\vee}|_{{{I}^{{\kern-0.74997pt}\wedge}}}:\sum_{i\in I}a_{i}\cdot
i\mapsto\sum_{j\in{{{J}^{{\kern-0.74997pt}\wedge}}}}\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}a_{i}Q_{j}(i){\operatorname{ev}}_{j},$

where ${\operatorname{ev}}_{j}$ is the dual basis, i.e.,
${\operatorname{ev}}_{j}(\delta_{j^{\prime}})=\delta_{jj^{\prime}}$, or
equivalently,

${\operatorname{ev}}_{j}:C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})\to{\mathbb{C}},f\mapsto
f(j).$

Note that the left is a finite sum, but the right may be an infinite sum
(which causes no problem, since the dual of a direct sum is the direct
product).

The following is a formal consequence:

$Q^{\vee}|_{{{I}^{{\kern-0.74997pt}\wedge}}}:\operatorname{avg}_{Y}^{2}\in{\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}\mapsto\operatorname{avg}_{Y}^{2}\in
C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})^{\vee}.$ (5.6)

###  5.2. Codes and designs

We define codes and designs. Let
$I_{C}\subset{{{I}^{{\kern-1.04996pt}\wedge}}}$ be a subset which does not
contain $i_{0}$ ($C$ for code). Let
$J_{D}\subset{{{J}^{{\kern-1.04996pt}\wedge}}}$ be a subset which does not
contain $j_{0}$ ($D$ for design). We define a convex cone
$(I_{C};J_{D})\subset{\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}$
by

$\displaystyle(I_{C};J_{D})$ $\displaystyle:=$ $\displaystyle\\{$
$\displaystyle\sum_{i\in I}a_{i}\cdot i\mid a_{i}=0\mbox{ for all but finite
$i$ },$ $\displaystyle a_{i}\geq 0\mbox{ for all
}i\in{{{I}^{{\kern-1.04996pt}\wedge}}},$ $\displaystyle a_{i}=0\mbox{ for all
}i\in I_{C},$
$\displaystyle\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}a_{i}Q_{j}(i)\geq
0\mbox{ for all }j\in{{{J}^{{\kern-1.04996pt}\wedge}}},$
$\displaystyle\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}a_{i}Q_{j}(i)=0\mbox{
for all }j\in J_{D}\ \ \\}.$

######  Definition 5.4.

A non-empty finite multi-subset $Y\subset{{{X}^{{\kern-1.04996pt}\wedge}}}$ is
called an $I_{C}$-free-code-$J_{D}$-design if $\operatorname{avg}_{Y}^{2}$
lies in the cone $(I_{C};J_{D})$. Furthermore, an $I_{C}$-free-
code-$\emptyset$-design [resp. an $\emptyset$-free-code-$J_{D}$-design] is
simply called an $I_{C}$-free-code [resp. a $J_{D}$-design].

More explicitly, the $i$-component of $\operatorname{avg}_{Y}^{2}$ in
${\mathbb{C}}^{\oplus{{{I}^{{\kern-0.74997pt}\wedge}}}}$ is

$a_{i}(Y):=\frac{1}{\\#Y^{2}}\\#((Y\times
Y)\cap{{{R}^{{\kern-1.04996pt}\wedge}}}^{-1}(i)),$ (5.7)

which is non-negative (5.2) and required to be $0$ for $i\in I_{C}$ (namely,
there is no pair $(x,y)\in Y\times Y$ with relation
${{{R}^{{\kern-1.04996pt}\wedge}}}(x,y)\in I_{C}$), and its
${\operatorname{ev}}_{j}$-component in
$C_{c}({{{J}^{{\kern-1.04996pt}\wedge}}})^{\vee}$ is

$b_{j}(Y):=\operatorname{avg}_{Y}^{2}(E_{j})=||\operatorname{avg}_{Y}^{j}||_{{\operatorname{HS}}}^{2},$
(5.8)

which is non-negative (5.5) and required to be $0$ for $j\in J_{D}$ (namely,
the $j$-component of $\operatorname{avg}_{Y}$ is zero for $j\in J_{D}$, or
equivalently, for any $f\in C({{{X}^{{\kern-1.04996pt}\wedge}}})_{j}$,
$\sum_{y\in Y}f(y)=0$ for $j\in J_{D}$). Note that $i_{0}$ is removed from
$I_{C}$, since

$\frac{1}{\\#Y^{2}}\\#((Y\times
Y)\cap{{{R}^{{\kern-1.04996pt}\wedge}}}^{-1}(i_{0}))=\frac{1}{\\#Y}$

is positive, and $j_{0}$ is removed since
$||\operatorname{avg}_{Y}^{j_{0}}||_{{\operatorname{HS}}}=1$ (being the
operator norm of the averaging for constant functions). Since

$\\#Y\operatorname{avg}_{Y}^{2}(i_{0})=1\mbox{ and
}\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}\\#Y\operatorname{avg}_{Y}^{2}(i)=\\#Y,$

we may consider an LP problem: under the constraint that
$(a_{i})_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}$ lies in the cone
$(I_{C};J_{D})$ and $a_{i_{0}}=1$, maximize/minimize
$\sum_{i\in{{{I}^{{\kern-0.74997pt}\wedge}}}}a_{i}$, which gives an