Semidefinite programming, harmonic analysis and coding theory

HAL Id: hal-00419745

https://hal.archives-ouvertes.fr/hal-00419745v2

Preprint submitted on 8 Sep 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Semidefinite programming, harmonic analysis and coding theory

Christine Bachoc

To cite this version:

Christine Bachoc. Semidefinite programming, harmonic analysis and coding theory. 2009. �hal- 00419745v2�

CODING THEORY

CHRISTINE BACHOC

ABSTRACT. These lecture notes where presented as a course of the CIMPA summer school in Manila, July 20-30, 2009, Semidefinite programming in al- gebraic combinatorics. This version is an update of June 2010.

CONTENTS

1. Introduction 2

1.1. Notations: 3

2. Linear representations of finite groups 3

2.1. Definitions 3

2.2. Examples 4

2.3. Irreducibility 5

2.4. The algebra ofG-endomorphisms 7

2.5. Orthogonality relations 7

2.6. Characters 8

2.7. Induced representation and Frobenius reciprocity 9

2.8. Examples from coding theory 11

3. Linear representations of compact groups 13

3.1. Finite dimensional representations 14

3.2. Peter Weyl theorem 14

3.3. Examples 16

4. Harmonic analysis of compact spaces 17

4.1. Commuting endomorphisms and zonal matrices. 17

4.2. Examples:G-symmetric spaces. 21

4.3. Positive definite functions and Bochner theorem 22 5. Explicit computations of the matricesE_k(x, y) 26

5.1. 2-point homogeneous spaces. 26

5.2. X ={1, . . . , q}under the action ofSq 26

5.3. Theq-Hamming space 26

5.4. Other symmetric spaces 29

5.5. Three cases with non trivial multiplicities 30 6. An SDP upper bound for codes from positive definite functions 38

6.1. The2-point homogeneous spaces 39

6.2. Symmetric spaces 40

6.3. Other spaces with true SDP bounds 41

Date: today.

1991 Mathematics Subject Classification. 52C17, 90C22.

Key words and phrases. spherical codes, kissing number, semidefinite programming, orthogonal polynomials.

1

7. Lov´asz theta 41

7.1. Introduction to Lov´asz theta number 41

7.2. Symmetrization and theq-gones 42

7.3. Relation with Delsarte bound and withm(X, δ) 44

8. Strengthening the LP bound for binary codes 44

References 46

1. INTRODUCTION

In coding theory, the so-called linear programming method, introduced by Phi- lippe Delsarte in the seventies [16] as proved to be a very powerful method to solve extremal problems. It was initially developed in the framework of association schemes and then extended to the family of2-point homogeneous spaces, including the compact real manifolds having this property (see [18], [24], [13, Chapter 9]).

Let us recall that a2-point homogeneous space is a metric space on which a group Gacts transitively, leaving the distancedinvariant, and such that, for(x, y)∈X², there existsg ∈Gsuch that(gx, gy) = (x^′, y^′)if and only ifd(x, y) =d(x^′, y^′).

The Hamming spaceH_nand the unit sphere of the Euclidean spaceSⁿ⁻¹are core examples of such spaces which play a major role in coding theory. To such a space is associated a sequence of orthogonal polynomials (P_k)_k≥0 such that, for allC⊂X,

X

(c,c^′)∈C²

P_k(d(c, c^′))≥0.

These inequalities can be understood as linear constraints on the distance distribu- tion of a code and are at the heart of the LP method.

The applications of this method to the study of codes and designs are numerous:

very good upper bounds for the number of elements of a code with given minimal distance can be obtained with this method, including a number of cases where this upper bound is tight and leads to a proof of optimality and uniqueness of certain codes, as well as to the best known asymptotic bounds (see [16], [30], [24], [13, Chapter 9], [28]).

In recent years, the development of the theory of error correcting codes has in- troduced many other spaces with interesting applications. To cite a few, codes over various alphabets associated to various weights, quantum codes, codes for the multi antenna systems of communications involving more complicated manifolds like the Grassmann spaces, have successively focused attention. For these spaces there was a need for a generalization of the classical framework of the linear pro- gramming method. This generalization was developed for some of these spaces, see [44], [45], [2], [37]. It turns out that in each of these cases, a certain sequence of orthogonal polynomials enters into play but unlike the classical cases, these polynomials are multivariate.

Another step was taken when A. Schrijver in [40] succeeded to improve the classical LP bounds for binary codes with the help of semidefinite programming.

To that end he exploited SDP constraints on triples of points rather than on pairs, arising from the analysis of the Terwilliger algebra of the Hamming scheme. His method was then adapted to the unit sphere [4] in the framework of the represen- tations of the orthogonal group. The heart of the method is to evidence matrices

Z_k(x, y, z)such that for allC ⊂X, X

(c,c^′,c^′′)∈C³

Z_k(c, c^′, c^′′)0.

Another motivation for the study of SDP constraints onk-tuples of points can be found in coding theory. It appears that not only functions on pairs of points such as a distance function d(x, y) are of interest, but also functions onk-tuples have relevant meaning, e.g. in connection with the notion of list decoding.

In these lecture notes we want to develop a general framework based on har- monic analysis of compact groups for these methods. In view of the effective ap- plications to coding theory, we give detailed computations in many cases. Special attention will be paid to the cases of the Hamming space and of the unit sphere.

Section 2 develops the basic tools needed in the theory of representations of fi- nite groups, section 3 is concerned with the representations of compact groups and Peter Weyl theorem. Section 4 discusses the needed notions of harmonic analy- sis: the zonal matrices are introduced and the invariant positive definite functions are characterized with Bochner theorem. Section 5 is devoted to explicit computa- tions of the zonal matrices. Section 6 shows how the determination of the invariant positive definite functions leads to an upper bound for codes with given minimal distance. Section 7 explains the connection with the so-called Lov´asz theta num- ber. Section 8 shows how SDP bounds can be used to strengthen the classical LP bounds, with the example of the Hamming space.

1.1. Notations: for a matrixAwith complex coefficients,A^∗stands for the trans- posed conjugate matrix. A squared matrix is said to be hermitian if A^∗ = Aand positive semidefinite if it is hermitian and all its eigenvalues are non negative. This property is denotedA0. We follow standard notations for sets of matrices: the set of n×mmatrices with coefficients in a fieldK is denotedK^n×m; the group of n×ninvertible matrices by Gl(Kⁿ); the group U(Cⁿ) of unitary matrices, respectivelyO(Rⁿ)of orthogonal matrices is the set of matricesA∈Gl(Cⁿ), re- spectively A ∈ Gl(Rⁿ)such thatA^∗ = A⁻¹. The space C^n×m is endowed with the standard inner product hA, Bi = Trace(AB^∗) = P

i,jA_i,jB_i,j. The number of elements of a finite setXis denotedcard(X)of|X|.

2. LINEAR REPRESENTATIONS OF FINITE GROUPS

In this section we shortly review the basic notions of group representation theory that will be needed later. There are many good references for this theory e.g. [41], or [38] which is mainly devoted to the symmetric group.

2.1. Definitions. LetGbe a finite group. A (complex linear) representation ofG is a finite dimensional complex vector spaceV together with a homomorphismρ:

ρ:G→Gl(V)

whereGl(V)is the general linear group ofV, i.e. the set of linear invertible trans- formations ofV. The degree of the representation (ρ, V)is by definition equal to the dimension ofV.

Two representations of Gsay(ρ, V) and (ρ^′, V^′) are said to be equivalent or isomorphic if there exists and isomorphismu:V →V^′ such that, for allg∈G,

ρ^′(g) =uρ(g)u⁻¹.

For example, the choice of a basis of V leads to a representation equivalent to (ρ, V)given by(ρ^′,C^d)whered= dim(V)andρ^′(g)is the matrix ofρ(g)in the chosen basis. In general, a representation ofGsuch thatV =C^dis called a matrix representation.

The notion of aG-module is equivalent to the above notion of representation and turns out to be very convenient. AG-module, or aG-space, is a finite dimensional complex vector spaceV such that for all g ∈ G, v ∈ V,gv ∈ V is well defined and satisfies the obvious properties: 1v=v,g(hv) = (gh)v,g(v+w) =gv+gw, g(λv) = λ(gv) forg, h ∈ G, v, w ∈ V,λ ∈ C. In other words,V is endowed with a structure of C[G]-module. One goes from one notion to the other by the identificationgv =ρ(g)(v). The notion of equivalent representations corresponds to the notion of isomorphic G-modules, an isomorphism ofG-modules being an isomorphism of vector spaces u : V → V^′ such that u(gv) = gu(v). Note that here the operations of Gon V and V^′ are denoted alike, which may cause some confusion.

2.2. Examples.

• The trivial representation1:V =Candgv=v.

• Permutation representations: letXbe a finite set on whichGacts (on the left). Let VX := ⊕x∈XCex. A natural action of G on VX is given by ge_x =e_gx, and defines a representation ofG, of degree|X|. The matrices of this representation (in the basis{e_x}) are permutation matrices.

– The symmetric group S_n acts on X = {1,2, . . . , n}. This action defines a representation of degreenofS_n.

– For allw,1≤w≤n,S_nacts on the setX_wof subsets of{1,2, . . . , n} of cardinal w. In coding theory an element of X_w is more likely viewed as a binary word of length n and Hamming weightw. The spacesX_w are called the Johnson spaces and denotedJ_n^w.

• The regular representation is obtained with the special case X = Gwith the action ofGby left multiplication. In the case G = S_n it has degree n!.. It turns out that the regular representation contains all building blocks of all representations ofG.

• Permutation representations again: ifGacts transitively onX, this action can be identified with the left action ofGon the left cosetsG/H ={gH : g∈G}whereH= Stab(x₀)is the stabilizer of a base point.

– The symmetric groupS_nacts transitively onX = {1,2, . . . , n}and the stabilizer of one point (sayn) can be identified with the symmetric groupS_n−1acting on{1, . . . , n−1}.

– The action ofS_nonJ_n^wis also transitive and the stabilizer of one point (say 1^w0^n−w) is the subgroup S_{1,...,w}×S{w+1,...,n} isomorphic to Sw×Sn−w.

– The Hamming spaceH_n={0,1}ⁿ =Fⁿ

2 affords the transitive action ofG= T⋊SnwhereT is the group of translationsT ={tu :u ∈ H_n},t_u(v) =u+vandS_npermutes the coordinates. The stabilizer of0ⁿis the group of permutationsS_n.

• Another way to see the permutation representations is the following: let C(X) :={f :X→C}

be the space of functions fromXtoC. The action ofGonX extends to a structure ofG-module onC(X)given by:

gf(x) :=f(g⁻¹x).

For the Dirac functions δ_y (δ_y(x) = 1 if x = y, 0 otherwise), the ac- tion ofGis given bygδ_y = δ_gy thus this representation is isomorphic to the permutation representation defined byX. This apparently more com- plicated presentation of permutation representations has the advantage to allow generalization to infinite groups acting on infinite spaces as we shall encounter later.

2.3. Irreducibility. LetV be aG-module (respectively a representation(ρ, V)of G). A subspaceW ⊂V is said to beG-invariant (orG-stable, or aG-submodule, or a subrepresentation of(ρ, V)), ifgw ∈ W (respectively ρ(g)(w) ∈W) for all g∈G,w∈W .

Example:V = V_G and W = Ce_G withe_G = P

g∈Ge_g. The restriction of the action ofGtoW is the trivial representation.

AG-moduleV is said to be irreducible if it does not contain any subspace W, W 6={0}, V, invariant underG. Otherwise it is called reducible.

Example: The G-invariant subspaces of dimension 1are necessarily irreducible.

If G is abelian, a G-module of dimension greater than 1 cannot be irreducible, because endomorphisms that pairwise commute afford a common basis of eigen- vectors.

The main result is then the decomposition of aG-module into the direct sum of irreducible submodules:

Theorem 2.1 (Maschke’s theorem). AnyG-module V 6={0}is the direct sum of irreducibleG-submodulesW₁, . . . , W_k:

(1) V =W₁⊕W₂⊕ · · · ⊕W_k.

Proof. By induction, it is enough to prove that anyG-submoduleW ofV affords a supplementary subspace which is alsoG-invariant. The main idea is to construct aG-invariant inner product and then prove that the orthogonal ofW for this inner product makes the job.

We start with an inner product hx, yi defined onV. There are plenty of them sinceV is a finite dimensional complex vector space. For example we can choose an arbitrary basis ofV and declare it to be orthonormal. Then we average this inner product onG, defining:

hx, yi^′ :=X

g∈G

hgx, gyi.

It is not difficult to check that we have defined a inner product which isG-invariant.

It is also easy to see that

W^⊥:={v∈V :hv, wi^′ = 0for allw∈W} isG-invariant, thus we have the decomposition ofG-modules:

V =W ⊕W^⊥

It is worth to notice that the above decomposition may not be unique. It is clear if one thinks of the extreme caseG={1}for which the irreducible subspaces are simply the one dimensional subspaces ofV. The decomposition ofV into the direct sum of subspaces of dimension1is certainly not unique (ifdim(V)>1of course).

But uniqueness is fully satisfied by the decomposition into isotypic subspaces. In order to define them we take the following notation: let R be a complete set of pairwise non isomorphic irreducible representations ofG. We have seen that any G-module affords aG-invariant inner product so the action ofGonRis expressed by unitary matrices in a given orthonormal matrix ofR. According to the context we viewR either as aG-module or as a homomorphismg 7→ R(g) ∈U(Cⁿ). It will turn out that there is only a finite number of them but we have not proved it yet.

The isotypic subspace IRofV associated toR ∈ Ris defined, with the notations of (1), by:

(2) IR:=⊕Wi≃RWi.

Theorem 2.2. LetR ∈ R. The isotypic spacesIRdo not depend on the decom- position ofV as the direct sum ofG-irreducible subspaces. We have the canonical decomposition

V =⊕R∈RIR.

Any G-subspace W ⊂ V such that W ≃ R is contained in IR and any G- irreducible subspace ofIRis isomorphic toR. A decomposition into irreducible subspaces ofIRhas the form

IR=W₁⊕ · · · ⊕W_mR

withW_i ≃R. Such a decomposition is not unique in general but the numberm_R does not depend on the decomposition and is called the multiplicity ofRinV.

Moreover, if V is endowed with aG-invariant inner product, then the isotypic spaces are pairwise orthogonal.

Proof. We start with a lemma which points out a very important property of irre- ducibleG-modules.

Lemma 2.3 (Schur Lemma). Let R₁ and R₂ two irreducible G-modules and let ϕ:R₁ → R₂ be aG-homomorphism. Then eitherϕ= 0orϕis an isomorphism ofG-modules.

Proof. The subspaces kerϕ and imϕare G-submodules of respectively R₁ and

R2thus they are equal to either{0}orRi.

We go back to the proof of the theorem. We start with the decomposition (1) ofV and the definition (2) of IR, a priori depending on the decomposition. Let W ⊂V, aG-submodule isomorphic toR. We apply Lemma 2.3 to the projections p_Wi and conclude that eitherp_Wi(W) ={0}orp_Wi(W) =W_iand this last case can only happen if W ≃ W_i. It proves thatW ⊂ IR and that aG-irreducible subspace ofIRcan only be isomorphic toR. It also proves that

IR= X

W⊂V,W≃R

W

hence giving a characterization of IR independent of the initial decomposition.

The numbermRmust satisfydim(IR) =mRdim(R)so it is independent of the decomposition ofIR.

IfV is equipped with aG-invariant inner product, we consider orthogonal pro- jections. Schur Lemma shows that P_W(W^′) = {0} or = W if W and W^′ are irreducible. Thus if they are notG-isomorphic,W andW^′must be orthogonal.

2.4. The algebra of G-endomorphisms. Let V be a G-module. The set of G- endomorphisms ofV is an algebra (for the laws of addition and composition) de- notedEnd_G(V). The next theorem describes the structure of this algebra.

Theorem 2.4. IfV ≃ ⊕R∈RR^mR, then End_G(V)≃ Y

R∈R

C^mR×mR.

Proof. The proof is in three steps: we shall assume firstV =Ris irreducible, then V ≃R^m, then the general case. Schur Lemma 2.3 is the main tool here.

IfV is irreducible, letϕ∈End_G(V). SinceV is a complex vector space,ϕhas got an eigenvalue λ. Thenϕ−λIdis aG-endomorphism with a non trivial kernel so from Schur Lemmaϕ−λId = 0. We have proved that

End_G(V) ={λId, λ∈C} ≃C.

We assume now thatV ≃R^mand we fix a decompositionV =W₁⊕ · · · ⊕W_m. For all 1 ≤i≤ j ≤ m, letuj,i :Wi → Wj an isomorphism ofG-modules such that the relations

u_k,j◦u_j,i =u_k,iandu_i,i = Id

hold for alli, j, k. Letϕ∈End_G(V); we associate toϕan element ofC^m×m in the following way. From previous discussion of the irreducible case it follows that for alli, jthere existsai,j ∈Csuch that, for allv∈Wi,

pWj◦ϕ(v) =aj,iuj,i(v).

The matrixA = (a_i,j)is the matrix associated toϕ. The proof that the mapping ϕ7→ Ais an isomorphism of algebras carries without difficulties and is left to the reader.

In the general case,V =⊕R∈RIR. Letϕ∈End_G(V). It is clear thatϕ(IR)⊂ IRthus

EndG(V) =⊕R∈REndG(IR)

and we are done.

It is worth to notice thatEnd_G(V)is a commutative algebra if and only if all the multiplicitiesm_Rare equal to either0or1. In this case we say thatV is multiplicity free. It is also the unique case when the decomposition into irreducible subspaces (1) is unique.

2.5. Orthogonality relations. Another important result which is a consequence of Schur lemma is the orthogonality relations between the matrix coefficients of the elements ofR:

Theorem 2.5. ForR∈ R, letd_R:= dim(R). For allR, S ∈ R,i, j, k, l, hR_i,j, S_k,li= 1

d_Rδ_R,Sδ_i,kδ_j,l.

Proof. ForA∈C^dR×dS, let A^′ = 1

|G| X

g∈G

R(g)AS(g)⁻¹.

This matrix satisfies R(g)A^′ = A^′S(g) for allg ∈ G. In other words it defines an homomorphism ofG-modules from(C^dS, S)to(C^dR, R). Schur lemma shows that if S 6= R, A^′ = 0and if S = R, A^′ = λId. Computing the trace of A^′ shows thatλ= Trace(A)/d_R. TakingA=E_i,j the elementary matrices, with the property thatS(g)⁻¹ =S(g)^∗, leads to the announced formula.

2.6. Characters. The character of a representation (ρ, V) ofG is the function χ_ρ:G→Cdefined by

χ_ρ(g) = Trace(ρ(g)).

As a consequence of the standard property of traces of matrices Trace(AB) = Trace(BA), the character of a representation only depends on its equivalence class, and it is a complex valued function onGwhich is constant on the conjugacy classes ofG(such a function is called a class function). The inner product of any twoχ, ψ∈ C(G)is defined by

hχ, ψi := 1

|G| X

g∈G

χ(g)ψ(g).

We have the very important orthogonality relations between characters:

Theorem 2.6 (Orthogonality relations of the first kind). Let χand χ^′ be respec- tively the characters of two irreducible representations (ρ, V)and (ρ^′, V^′)of G.

Then

hχ, χ^′i=

(1 ifρ≃ρ^′ 0 otherwise.

Proof. It is a straight forward consequence of Theorem 2.5, since the trace of a representation is the sum of the diagonal elements of any equivalent matrix repre-

sentation.

A straightforward consequence of the above theorem is that hχ_ρ, χ_Ri = m_R for all R ∈ R. This property is a very convenient tool to study the irreducible decomposition of a given representation (ρ, V) ofG; in particular it shows that a representation is irreducible if and only if its character χsatisfieshχ, χi = 1. In the case of the regular representation it leads to the following very important result:

Theorem 2.7. [Decomposition of the regular representation]

C(G) ≃ ⊕R∈RR^dim(R)

Proof. Compute the character of the regular representation.

A consequence of the above theorem is the finiteness of the number of irre- ducible representations of a given finite group, together with the formula

|G|= X

R∈R

(dim(R))²

which shows e.g. completeness of a given set of irreducibleG-modules.

A second consequence of the orthogonality relations is that a representation of Gis uniquely characterized up to isomorphism by its character.

Theorem 2.8.

(ρ, V)≃(ρ^′, V^′)⇐⇒χ_ρ=χ_ρ^′.

Proof. Ifχ_ρ=χ_ρ′, the multiplicities of an irreducible representation ofGare the

same inV andV^′, henceV ≃GV^′.

Let us denote byR(G)the subspace of elements ofC(G)which are constant on the conjugacy classesC1, . . . , Cs ofG. The dimension ofR(G) is obviously the numbersof conjugacy classes of G. We have seen that the charactersχ_Rof the irreducible representations ofGbelong toR(G)and form an orthonormal family.

It turns out that they in fact form a basis of R(G), which in other words means that the number of irreducible representations of Gis exactly equal to its number of conjugacy classes.

Theorem 2.9.

R(G) =⊕R∈RCχR.

Proof. It is clear thatC(G) =C[G]δ_e. ThusEnd_G(C(G)) ≃C[G]. In particular, the center ofEnd_G(C(G))is isomorphic to the centerZ(C[G])ofC[G]. It is easy to verify that the center ofC[G]is the vector space spanned by the elementsλ_i :=

P

g∈Cigassociated to each conjugacy classC_iofG, thusZ(C[G])is of dimension sthe number of conjugacy classes ofG. On the other hand, as a consequence of Theorem 2.7 and Theorem 2.4, we have End_G(C(G)) ≃ Q

R∈RC^dR×dR where d_R = dim(R). Thus the center of End_G(C(G)) is isomorphic to C^|R| and we have proved that the number of G-irreducible modules is equal to the number of

conjugacy classes ofG.

Remark 2.10. There is not in general a natural bijection between the set of con- jugacy classes ofGand the set of its irreducible representations. However, in the special case of the symmetric groupSn, such a correspondance exists. The conju- gacy classes are naturally indexed by the partitionsλofnand to every partitionλ ofnis associated an irreducible moduleS^λalso called a Specht module (see [38]).

2.7. Induced representation and Frobenius reciprocity. Induction is a way to construct representations of a groupGfrom representations of its subgroups. Look- ing at the irreducible subspaces of representations that are induced from subgroups is a very convenient way to find new irreducible representations of a group G. In- duction is an operation on representations which is dual to the easier to understand restriction. IfV is aG-module andHis a subgroup ofG, the restrictionRes^G_H(V) is simply the space V considered as a C[H]-module. If V is an H-module, we defineInd^G_H(V)to be theC[G]-module

Ind^G_H(V) :=C[G]⊗^C[H]V.

Here we exploit the bi-module structure of C[G](the tensor product over C[H]

means that λµ⊗v =λ⊗µvwhenµ∈C[H]). A more explicit (but less intrisic) formulation for this construction is the following: let{x₁, . . . , x_t}be a complete system of representatives ofG/H, so thatG=x₁H∪ · · · ∪x_tH. Then

Ind^G_H(V) =⊕^ti=1x_iV

where the left action of G is as follows: for all i, there is j and h ∈ H both depending on gsuch thatgx_i = x_jh. Then gx_iv := x_j(hv) wherehv ∈ V. A

third construction ofInd^G_H(V)is the following:

Ind^G_H(V) ={f :G→V such thatf(gh) =h⁻¹f(g)}. The equivalence of these three formulations is a recommended exercise !

Example: The permutation representation ofGonX =G/H is nothing else than the induction of the trivial representation ofH. In short,C(X) = Ind^G_H1.

Since the induction of two isomorphicH-modules leads to isomorphicG-modules and similarly for the restriction, these operations act on the characters thus we de- note similarlyRes^G_Hχ,Ind^G_Hχthe characters of the corresponding modules.

Lemma 2.11. Letχbe a character ofH. The induced characterInd^G_Hχis given by the formula:

Ind^G_Hχ(g) = 1

|H| X

x∈G x⁻¹gx∈H

χ(x⁻¹gx).

Proof. We take a decompositionInd^G_H(V) =x₁V⊕· · ·⊕x_tV where{x₁, . . . , x_t} are representatives ofG/H. Sincegx_iv=x_jhvwith the notations above,g(x_iV)⊂ x_jV and the blockx_iV will contribute to the trace ofx 7→ gxonly whenj = i, which corresponds to the case whenx⁻¹_i gx_i ∈ H. Then, the multiplication byg onx_iV acts like the multiplication byh=x⁻¹_i gx_i on V. Thus we have

Ind^G_Hχ(g) = X

1≤i≤t x⁻¹_i gxi∈H

χ(x⁻¹_i gx_i)

= 1

|H| X

x∈G x⁻¹gx∈H

χ(x⁻¹gx).

The duality between the operations of restriction and induction is expressed in the following important theorem:

Theorem 2.12 (Frobenius reciprocity). LetHbe a subgroup ofGand letχandψ be respectively a character ofHand a character ofG. Then

hInd^G_Hχ, ψi=hχ,Res^G_Hψi.

Proof. Letχ˜ : G → C be defined by: χ(g) =˜ χ(g) if g ∈ H andχ(g) = 0˜ if g /∈H(of courseχ˜is not a character ofG). We computehInd^G_Hχ, ψi:

hInd^G_Hχ, ψi= 1

|G| X

x∈G

Ind^G_Hχ(g)ψ(g)

= 1

|G||H| X

g∈G

X

x∈G

˜

χ(x⁻¹gx) ψ(g)

= 1

|G||H| X

x∈G

X

g∈G

˜

χ(x⁻¹gx)ψ(g)

= 1

|G||H| X

x∈G

X

g^′∈G

˜

χ(g^′)ψ(xg^′x⁻¹)

= 1

|G||H| X

x∈G

X

g^′∈G

˜

χ(g^′)ψ(g^′)

= 1

|H| X

h∈H

χ(h)ψ(h) =hχ,Res^G_Hψi.

2.8. Examples from coding theory. In coding theory we are mostly interested in the decomposition of C(X)under the action of G = Aut(X) for various spaces X. We recall that the action ofGonf ∈ C(X)is given by(gf)(x) =f(g⁻¹x).

The spaceC(X)is endowed with the inner product hf, f^′i= 1

|X| X

x∈X

f(x)f^′(x).

which isG-invariant.

2.8.1. The binary Hamming spaceH_n: recall thatG=T⋊S_n. Let, fory∈H_n, χ_y ∈ C(H_n)be defined by χ_y(x) = (−1)^x·y. The set{χ_y, y ∈ H_n}is exactly the set of irreducible characters of the additive groupFⁿ

2, and form an orthonormal basis ofC(H_n). The computation of the action ofGonχ_y shows that forσ ∈S_n, σχ_y =χ_σ(y)and fort_u ∈T,t_uχ_y = (−1)^u·yχ_y. Let, for0≤k≤n,

P_k :=⊥y,wt(y)=k Cχy

Thus P_k is aG-invariant subspace ofC(H_n) of dimension ⁿ_k

and we have the decomposition

(3) C(H_n) =P₀ ⊥P₁⊥ · · · ⊥P_n.

The computation hχPk, χPki = 1whereχPk is the character of theG-moduleP_k shows that these modules areG-irreducible.

Now we introduce the Krawtchouk polynomials. The elementZ_k:=P

wt(y)=kχ_y ofC(H_n)isS_n-invariant. In other words,Z_k(x)only depends onwt(x). We define

the Krawtchouk polynomialK_kfor0≤k≤nby K_k(w) : =Z_k(x) = X

wt(y)=k

(−1)^x·ywherewt(x) =w (4)

=

w

X

i=0

(−1)ⁱ w

i

n−w k−i

. (5)

We review some properties of these polynomials:

(1) deg(K_k) =k (2) K_k(0) = ⁿ_k

(3) Orthogonality relations: for all0≤k≤l≤n 1

2ⁿ

n

X

w=0

n w

K_k(w)K_l(w) =δ_k,l n

k

The last property is just a reformulation of the orthogonality of theZ_k∈P_k, since, iff, f^′∈ C(Hn)areSn-invariant, andf˜(w) :=f(x),wt(x) =w,

hf, f^′i= 1 2ⁿ

X

x∈Hn

f(x)f^′(x)

= 1 2ⁿ

n

X

w=0

n w

f˜(w) ˜f^′(w).

The above three properties characterize uniquely the Krawtchouk polynomials.

LetC ⊂H_nbe a binary code. Let1C be the characteristic function ofC. The obvious inequalities hold:

(6) 0≤k≤n, X

wt(y)=k

h1C, χ_yi² ≥0.

Since the decomposition of1Cover the basisχ_y reads 1C = X

y∈Hn

h1C, χ_yiχ_y.

the above inequalities are indeed reformulations of the non negativity of the squared norm of the projectionsp_Pk(1C). They express in terms of the Krawtchouk poly- nomials:

(7) 0≤k≤n, 1

2²ⁿ X

(x,x^′)∈C²

K_k(d_H(x, x^′))≥0 or equivalently in terms of the distance distribution of the codeC: if

A_w(C) := 1

|C||{(x, x^′)∈C² :d_H(x, x^′) =w}|

then

0≤k≤n, |C| 2²ⁿ

n

X

w=0

A_w(C)K_k(w)≥0.

These inequalities are the basic inequalities involved in Delsarte linear program- ming method. We shall encounter similar inequalities in a very general setting.

In the special case whenCis linear, we have h1C, χ_yi= |C|

2ⁿ 1C^⊥(y) so that we recognise the identity

X

wt(y)=k

h1C, χyi² = |C| 2²ⁿ

n

X

w=0

Aw(C)K_k(w) to be the Mac Williams identity

A_k(C^⊥) = 1

|C|

n

X

w=0

A_w(C)K_k(w).

The more general q-ary Hamming space affords similar results; it is treated in 5.3.

2.8.2. The Johnson spacesJ_n^w: the group isG=S_n. Here, we shall see at work a standard way to evidence G-submodules as kernels of G-endomorphisms. For details we refer to [17] where theq-Johnson spaces are given a uniform treatment.

We introduce the applications

δ :C(J_n^w)→ C(J_n^w−1)

f 7→δ(f) :δ(f)(x) := X

y∈J_n^w, x⊂y

f(y) and

ψ:C(J_n^w−1)→ C(J_n^w)

f 7→ψ(f) :ψ(f)(x) := X

y∈Jn^w−1, y⊂x

f(y)

Both of these applications commute with the action ofG. They satisfy the follow- ing properties: hf, ψ(f^′)i=hδ(f), f^′i,ψis injective andδis surjective. Therefore the subspace ofC(J_n^w):

H_w:= kerδ is aG-submodule of dimension _wⁿ

− _w−1ⁿ

and we have the orthogonal decom- position

C(J_n^w) =H_w⊥ψ(C(J_n^w−1))≃H_w ⊥ C(J_n^w−1).

By induction we obtain a decomposition

C(J_n^w)≃H_w ⊥H_w−1 ⊥ · · · ⊥H₀

which can be proved to be the irreducible decomposition ofC(J_n^w)(see 5.3.1).

3. LINEAR REPRESENTATIONS OF COMPACT GROUPS

In this section we enlarge the discussion to the representation theory of compact groups. For this section we refer to [12].

3.1. Finite dimensional representations. The theory of finite dimensional repre- sentations of finite groups extends nicely and straightforwardly to compact groups.

A finite dimensional representation of a compact group Gis a continuous homo- morphismρ:G→Gl(V)whereV is a complex vector space of finite dimension.

A compact group G affords a Haar measure, which is a regular left and right invariant measure. We assume this measure to be normalized, i.e. the groupGhas measure1. With this measure the finite sums over elements of a finite group can be replaced with integrals; so the crucial construction of aG-invariant inner product in the proof of Maschke theorem extends to compact groups with the formula

hx, yi^′ :=

Z

Ghgx, gyidg.

Hence Maschke theorem remains valid for finite dimensional representations. We keep the notationRfor a set of representatives of the finite dimensional irreducible representations of G, chosen to be representations with unitary matrices. A main difference with the finite case is thatRis not finite anymore.

3.2. Peter Weyl theorem. Infinite dimensional representations immediately oc- cur with the generalization of permutation representations. Indeed, ifGacts con- tinuously on a spaceX, it is natural to consider the action ofGon the spaceC(X) of complex valued continuous functions onXgiven by(gf)(x) =f(g⁻¹x)to be a natural generalization of permutation representations. A typical example of great interest in coding theory is the action of G = O(Rⁿ) on the unit sphere of the Euclidean space:

Sⁿ⁻¹:={x∈Rⁿ : x·x= 1}.

The regular representation, which is the special caseC(G), with the left action of Gon itself, can be expected to play an important role similar to the finite case. It is endowed with the inner product

hf, f^′i:=

Z

G

f(g)f^′(g)dg.

ForR ∈ R, the matrix coefficientsg → R_i,j(g)belong to unitary matrices. The- orem 2.5 establishing the orthogonality relations between the matrix coefficients of the elements ofRremains valid; thus they form an orthogonal system inC(G).

The celebrated Peter Weyl theorem asserts that these elements span a vector space which is dense inC(G)for the topology of uniform convergence.

Theorem 3.1. [Peter Weyl theorem] The finite linear combinations of the functions R_i,jare dense inC(G)for the topology of uniform convergence.

Proof. We give a sketch of the proof:

(1) IfV is a finite dimensional subspace ofC(V)which is stable by right trans- lation (i.e. bygf(x) =f(xg)) andf ∈V, thenf is a linear combination of a finite number of theR_i,j: according to previous discussion, there is a decomposition V = W₁ ⊕ · · · ⊕W_n such thatW_k is irreducible. If W_k ≃R, there exists a basise₁, . . . , e_dR ofW_kin which the action ofG has matricesR. Explicitly,

e_j(hg) =

dR

X

i=1

R_i,j(g)e_i(h).

Takingh= 1, we obtaine_j =PdR

i=1e_i(1)R_i,j.

(2) The idea is to approximate f ∈ C(G) by elements of such subspaces, constructed from the eigenspaces of a compact selfadjoint operator. We introduce the convolution operators: letφ∈ C(G),

T_φ(f)(g) = (φ∗f)(g) = Z

G

φ(gh⁻¹)f(h)dh.

(3) Since G is compact, f is uniformly continuous; this property allows to chooseφsuch thatkf −T_φ(f)k∞is arbitrary small.

(4) The operator T_φ is compact and can be assumed to be selfadjoint. The spectral theorem for such operators on Hilbert spaces (hereL²(G)) asserts that the eigenspaces V_λ := {f :T_φf = λf}forλ 6= 0are finite dimen- sional and that the space is the direct Hilbert sum⊕λV_λ. Fort > 0, the subspacesV_t:=⊕V_λ,|λ|>thave finite dimension (i.e. there is only a finite number of eigenvaluesλwith|λ|> t >0).

(5) The operatorT_φcommutes with the action of Gby right translation thus the subspacesV_λ are stable under this action.

(6) Let f_λ be the projection of f on V_λ. The finite sums f_t := P

|λ|>tf_λ converge tof−f₀for theL²-norm whent→0.

(7) Moreover, for all f ∈ C(V), kT_φ(f)k∞ ≤ kφk∞kfk2. Thus, T_φ(ft) converges uniformly toT_φ(f−f₀) =T_φ(f). Finally,T_φ(f_t)∈V_tandV_tis finite dimensional and invariant under the action ofGby right translations, thus by (1)T_φ(f_t)is a linear combinations of theR_i,j.

If d_R = dim(R), the vector space spanned by {R_i,j, i = 1, . . . , d_R} is G- invariant and isomorphic toR. So Peter-Weyl theorem means that the decomposi- tion of the regular decomposition is

C(G) =⊥R∈R IR

where IR ≃ R^dR, generalizing Theorem 2.7 (one has a better understanding of this decomposition with the action ofG×GonGgiven by(g, g^′)h=ghg^′−1. For this action C(G) = ⊕R∈RR⊗R^∗ whereR^∗ is the contragredient representation, andR⊗R^∗ isG×G-irreducible).

Since uniform convergence is stronger than L² convergence, we also have as a consequence of Peter Weyl theorem that the matrix coefficients R_i,j (suitably rescaled) form an orthonormal basis ofL²(G)in the sense of Hilbert spaces.

A slightly more general version of Peter Weyl theorem deals with the decom- position ofC(X) whereX is a compact space on which Gacts homogeneously.

IfG_x0 is the stabilizer of a base pointx₀ ∈X, thenX can be identified with the quotient spaceG/G_x0. The Haar measure onGgives rise to aG-invariant regular measureµonXandC(X)is endowed with the inner product

hf, f^′i:= 1 µ(X)

Z

X

f(x)f^′(x)dµ(x).

The spaceC(X)can be identified with the spaceC(G)^Gx0 ofG_x0-invariant (for the right translation) functions thusC(X)affords a decomposition of the form

C(X)≃⊥R∈RR^mR

for some integers m_R, 0 ≤ m_R ≤ d_R, in the sense of uniform as well as L² convergence.

A more serious generalization of the above theorem deals with the unitary rep- resentations ofG. These are the continuous homomorphisms fromGto the unitary group of a Hilbert space.

Theorem 3.2. Let π : G → U(H) be a continuous homomorphism from G to the unitary group of a Hilbert space H. ThenH is a direct Hilbert sum of finite dimensional irreducibleG-modules.

Proof. The idea is to construct in H aG-subspace of finite dimension and then to iterate with the orthogonal complement of this subspace. To that end, for a fixed v ∈ H, one chooses f ∈ C(G) such that R

Gf(g)(π(g)v)dg 6= 0. From Peter Weyl theorem, f can be assumed to be a finite linear combination of the R_i,j. In other words, there exists a finite dimensional unitary representation(ρ, V)and e₁, e₂ ∈V such thatf(g) = hρ(g⁻¹)e₁, e₂iV. The operatorT :V → Hdefined by

T(x) = Z

Ghρ(g⁻¹)x, e₂iV(π(g)v)dg

commutes with the actions of G and is non zero. Thus its image is a non zero G-subspace of finite dimension ofH.

3.3. Examples.

3.3.1. The unit sphereSⁿ⁻¹: it is the basic example. The orthogonal groupG= O(Rⁿ) acts homogeneously onSⁿ⁻¹. The stabilizer G_x0 ofx₀ can be identified withO(x^⊥₀) ≃O(Rⁿ⁻¹). Hereµ= ωis the Lebesgue measure onSⁿ⁻¹. We set ω_n:=ω(Sⁿ⁻¹). The irreducible decomposition ofC(Sⁿ⁻¹)is as follows:

C(Sⁿ⁻¹) =H₀ⁿ⊥H₁ⁿ⊥. . . H_kⁿ⊥. . .

whereH_kⁿis isomorphic to the spaceHarmⁿ_k of harmonic polynomials:

Harmⁿ_k :={P ∈C[X₁, . . . , X_n]_k: ∆P = 0,∆ =

n

X

i=1

∂²

∂x²_i}

The spaceHarmⁿ_k is aO(Rⁿ)-module because the Laplace operator∆commutes with the action of the orthogonal group and it is moreover irreducible. Its dimen- sion equalshⁿ_k := ^n+k−1_k

− ^n+k−3_k−2

. The embedding ofHarmⁿ_kintoC(Sⁿ⁻¹)is the obvious one, to the corresponding polynomial function in thencoordinates.

3.3.2. The action of stabilizers of many points: for our purposes we are inter- ested in the decomposition of some spacesC(X), Xhomogeneous forG, for the action of a subgroupHofG, typicallyH =G_x1,...,xsthe stabilizer ofspoints. In order to describe it, it is enough to study the decomposition of the G-irreducible submodules of C(X) under the action ofH; thus we have to decompose only fi- nite dimensional spaces. However, because the same irreducible representation of H may occur in infinitely many of the G-isotypic subspaces, it happens that theH-isotypic subspaces are not of finite dimension. A typical example is given by X = Sⁿ⁻¹, G = O(Rⁿ) and H = G_e ≃ O(Rⁿ⁻¹). It is a classical result

that for the restricted action toHthe decomposition ofHarmⁿ_k intoH-irreducible subspaces is given by:

(8) Harmⁿ_k ≃

k

M

i=0

Harmⁿ⁻¹_i . Hence, each of theH_kⁿin (3.3.1) decomposes likewise:

H_kⁿ=H_0,kⁿ ⊥H_1,kⁿ ⊥. . .⊥H_k,kⁿ

where H_i,kⁿ ≃ Harmⁿ⁻¹_i . We have the following picture, where the H-isotypic components appear to be the rows of the second decomposition.

C(Sⁿ⁻¹) =_G H₀ⁿ ⊥ H₁ⁿ ⊥ . . . ⊥ H_kⁿ ⊥. . .

=H H_0,0ⁿ ⊥ H_0,1ⁿ ⊥ . . . ⊥ H_0,kⁿ ⊥. . .

⊥ H_1,1ⁿ ⊥ . . . ⊥ H_1,kⁿ ⊥. . .

· · · ·

⊥ H_k,kⁿ ⊥. . . 4. HARMONIC ANALYSIS OF COMPACT SPACES

We take notations for the rest of the lecture notes. Xis a compact space (possi- bly finite) on which a compact group (possibly finite)Gacts continuously. More- over, X is endowed with aG-invariant Borel regular measureµfor whichµ(X) is finite. IfXitself is finite, the topology is the discrete topology and the measure is the counting measure. In the previous sections we have discussed the decom- position of the permutation representation C(X). In order to lighten the notations, we assume that Ghas a countable number of finite dimensional irreducible rep- resentations (it is the case if G is a group of matrices over the reals since then L²(G)is a separable Hilbert space), and we let R = {R_k, k ≥ 0}, where R₀ is the trivial representation. We let d_k := dim(R_k). From Theorem 3.2, we have a decomposition

(9) C(X)⊂L²(X) =⊕k≥0,1≤i≤mkH_k,i

whereH_k,i⊂ C(X),H_k,i≃R_k,0≤m_k≤+∞(the casem_k = 0means thatR_k does not occur, the casem_k = +∞may occur if Gis not transitive onX). The isotypic subspaces are pairwise orthogonal and denotedIk:

Ik=⊕^m_i=1^kH_k,i

We take the subspacesH_k,ito be also pairwise orthogonal. For allk, i, we choose an orthonormal basise_k,i,1, . . . , e_k,i,dk ofH_k,isuch that in this basis the action of g∈Gis expressed by the unitary matrixR_k(g). The set{e_k,i,s}is an orthonormal basis in the Hilbert sense.

4.1. Commuting endomorphisms and zonal matrices. In this subsection we want to give more information on the algebra End_G(C(X)) of commuting con- tinuous endomorphisms of C(X). We introduce, for K ∈ C(X²), the operators TK, called Hilbert-Schmidt operators:

TK(f)(x) = 1 µ(X)

Z

X

K(x, y)f(y)dµ(y).

It is easy to verify thatTK ∈EndG(C(X))ifKisG-invariant, i.e. ifK(gx, gy) = K(x, y) for all g ∈ G, (x, y) ∈ X². A continuous function K(x, y) with this

property is also called a zonal function. It is also easy, but worth to notice that T_K◦T_K^′ =T_K∗K^′whereK∗K^′is the convolution ofK andK^′:

(K∗K^′)(x, y) :=

Z

X

K(x, z)K^′(z, y)dµ(z).

Let

K:={K∈ C(X²) :K(gx, gy) =K(x, y)for allg∈G,(x, y)∈X²}. The triple(K,+,∗)is aC-algebra (indeed aC^∗-algebra, withK^∗(x, y) :=K(y, x)).

Thus we have an embeddingK →End_G(C(X)).

AssumeV ⊂ C(X) is a finite dimensionalG-subspace such that V = W₁ ⊥

· · · ⊥ W_m with W_i ≃ R ∈ R. By the same proof as the one of Theorem 2.4, End_G(V) ≃ C^m×m. More precisely, we have seen that, if u_j,i : W_i → W_j are G-isomorphisms, such that u_k,j ◦u_j,i = u_k,i, then an element φ ∈ End_G(V) is associated to a matrix A = (a_i,j) ∈ C^m×m such that, for all f ∈ V, with p_Wi(f) =f_i,

φ(f) =

m

X

i,j=1

a_j,iu_j,i(f_i).

For all1≤i≤m, let(e_i,1, . . . , e_i,d),d= dim(R), be an orthonormal basis ofW_i such that in this basis the action ofg∈Gis expressed by the unitary matrixR(g).

We define

Ei,j(x, y) :=

d

X

s=1

ei,s(x)ej,s(y).

Then we have:

Lemma 4.1. The above defined functionsE_i,j satisfy:

(1) E_i,j is zonal:E_i,j(gx, gy) =E_i,j(x, y).

(2) LetTi,j :=TEi,j. ThenTj,i(Wi) =Wj andTj,i(W_k) = 0fork6=i.

(3) T_i,j◦T_j,k =T_i,k.

Proof. (1) From the construction, we have e_i,s(gx) =

d

X

t=1

R_s,t(g)e_i,t(x) thus

Ei,j(gx, gy) =

d

X

s=1

ei,s(gx)ej,s(gy)

=

d

X

s=1 d

X

k,l=1

R_s,k(g)R_s,l(g)e_i,k(x)e_j,l(y)

=

d

X

k,l=1

X^d

s=1

R_s,k(g)R_s,l(g)

e_i,k(x)e_j,l(y)

=

d

X

k

e_i,k(x)e_j,k(y) =E_i,j(x, y)

where the second last equality holds becauseR(g)is a unitary matrix.

(2) We computeT_j,i(e_k,t):

T_j,i(e_k,t)(x) = 1 µ(X)

Z

X d

X

s=1

e_j,s(x)e_i,s(y)

e_k,t(y)dµ(y)

= 1

µ(X)

d

X

s=1

e_j,s(x) Z

X

e_i,s(y)e_k,t(y)dµ(y)

=

d

X

s=1

e_j,s(x)he_k,t, e_i,si

=

d

X

s=1

e_j,s(x)δ_k,iδ_t,s=δ_k,ie_j,t(x).

(3) Similarly one computes that

Ei,j∗E_l,k=δ_j,lE_i,k.

The E_i,j(x, y)put together form a matrixE =E(x, y), that we call the zonal matrix associated to theG-subspaceV:

(10) E(x, y) := E_i,j(x, y)

1≤i,j≤m.

At this stage is is natural to discuss the dependence of this matrix on the various ingredients needed for its definition.

Lemma 4.2. We have

(1) E(x, y)is unchanged if another orthonormal basis ofWiis chosen (i.e. if another unitary representative of the irreducible representation Ris cho- sen).

(2) E(x, y)is changed toAE(x, y)A^∗for some matrixA∈Gl(C^m)if another decomposition (not necessarily with orthogonal spaces)V =W₁^′⊕ · · · ⊕ W_m^′ is chosen.

Proof. (1) Let (e^′_i,1, . . . , e^′_i,d) be another orthonormal basis ofWi and letUi

be unitaryd×dmatrices such that

(e^′_i,1, . . . , e^′_i,d) = (e_i,1, . . . , e_i,d)U_i.

Since we want the representationRto be realized by the same matrices in the basis(e^′_i,1, . . . , e^′_i,d)whenivaries, we haveU_i =U_j =U. Then, with obvious notations,

E_i,j^′ (x, y) =(e^′_i,1(x), . . . , e^′_i,d(x))(e^′_i,1(y), . . . , e^′_i,d(y))^∗

=(e_i,1(x), . . . , e_i,d(x))U U^∗(e_i,1(y), . . . , e_i,d(y))^∗

=(e_i,1(x), . . . , e_i,d(x))(e_i,1(y), . . . , e_i,d(y))^∗

=E_i,j(x, y).

(2) IfV = W₁ ⊥ · · · ⊥ W_m =W₁^′ ⊥ · · · ⊥ W_m^′ with basis(e_i,1, . . . , e_i,d) ofW_i and(e^′_i,1, . . . , e^′_i,d) ofW_i^′ in which the action ofGis by the same matrices R(g), let φ ∈ End(V) be defined by φ(e_i,s) = e^′_i,s. Clearly

φ commutes with the action of G; if u_j,i is defined byu_j,i(e_i,s) = e_j,s then we have seen that, for some matrix A = (a_i,j), e^′_i,s = φ(e_i,s) = P_m

j=1a_j,ie_j,s. MoreoverAis invertible. It is unitary if the spacesW_i^′ are pairwise orthogonal. With the notationsE(x) := (e_i,s(x)), we have

E(x, y) =E(x)E(y)^∗andE^′(x) =A^tE(x) thus

E^′(x, y) =A^tE(x, y)A.

Going back toφ∈End_G(V), from Lemma 4.1 we can takeu_j,i =T_j,iand we have the expression

φ=

m

X

i,j=1

a_j,iT_j,i=T_hA,Ei.

We take the following notation: the space of linear combinations of elements of the formf(x)g(y)for(f, g)∈V²is denotedV⁽²⁾. We have proved the following:

Proposition 4.3. Let

KV :={K∈V⁽²⁾ :K(gx, gy) =K(x, y)for allg∈G,(x, y)∈X²}. The following are isomorphisms ofC-algebras:

KV →EndG(V) C^m×m →EndG(V)

K 7→T_K A 7→T_hA,Ei.

Moreover,End_G(C(X))is commutative iffKis commutative iffm_k = 0,1for all k≥0.

Proof. The isomorphisms are clear from previous discussion. For the last assertion, it is enough to point out that

End_G(C(X)) = Y

k≥0

End_G(Ik).

Remark 4.4. Proposition 4.3 shows in particular that KV and End_G(V) have the same dimension. It is sometimes easy to calculate the dimension of KV; for example if X is a finite set and V = C(X), then dim(KV) is exactly equal to the number of orbits of G acting on X². On the other hand, in this case, the dimension of End_G(V) is the sum of the squares of the multiplicities inC(X) of the irreducible representations of G. For the binary Hamming space treated in 2.8.1, the orbits of G acting on X² are in one to one correspondance with the values taken by the Hamming distance, i.e. there are (n+ 1)such orbits. Thus, once we have obtained the decomposition C(H_n) =P₀ ⊥ · · · ⊥P_n, because this decomposition involves allready (n+ 1)subspaces, we can conclude readily that these subspaces are irreducible. This reasoning applies also to the Johnson space 2.8.2 and to the more generalq-Hamming space 5.3. A variant of this method is as follows: if we suspectV ⊂ C(X)to be irreducible, then it is enough to prove that KV has dimension1. See in 5.3.1 for an illustration.