Block diagonalization for algebra's associated with block codes
aa r X i v : . [ m a t h . O C ] O c t Block diagonalization for algebra’s associated with block codes
Dion Gijswijt ∗ October 26, 2018
Abstract
For a matrix ∗ -algebra B ⊆ C m × m , denote by A := Sym n ( B ) the matrix ∗ -algebraconsisting of the elements in the n -fold tensor product B ⊗ n that are invariant under per-muting the n factors in the tensor product. Examples of such algebras in coding theoryinclude the Bose-Mesner algebra and Terwilliger algebra of the (non)binary Hammingcube and algebras arising in SDP-hierarchies for codes using moment matrices. We givea computationally efficient block diagonalization of A in terms of a given block diago-nalization of B . As a tool we use some basic facts from the representation theory of thesymmetric group. Keywords: block diagonalization, semidefinite programming, Terwilliger algebra, as-sociation scheme, representation theory, symmetric group.
A matrix ∗ -algebra is a set A ⊆ C n × n of matrices that is closed under addition, scalarmultiplication, matrix multiplication and A A ∗ (taking the conjugate transpose). It is aclassical result that any such algebra can be brought into block diagonal form. That is, thereexists an isomorphism φ : A → t M i =1 C p i × p i (1)of ∗ -algebras, meaning that φ is a linear bijection, φ ( AB ) = φ ( A ) φ ( B ) and φ ( A ∗ ) = φ ( A ) ∗ for all A, B ∈ A .In this paper we will be concerned with constructing in a computationally efficient waysuch block diagonalizations φ . The motivation comes from semidefinite programming, whereblock diagonalization has proven to be a valuable tool since it can be used to reduce thecomplexity of semidefinite programs having a large group of symmetries.Semidefinite programming is an extension of linear programming that is both very generaland at the same time can be performed efficiently (up to given precision), both in theory andin practice. A (complex) semidefinite program is an optimization problem of the formmaximize h C, X i subject to X (cid:23) h A i , X i = b i for i = 1 , . . . , m , (2) ∗ CWI and University of Leiden. Mailing address: CWI, Science Park 123, 1098 XG Amsterdam, TheNetherlands. Email: [email protected]. X ∈ C n × n is an Hermitian matrix variable, C, A , . . . , A m ∈ C n × n are given Hermitianmatrices and b , . . . , b m are given real numbers. Here X (cid:23) X is positivesemidefinite and h C, X i := trace X ∗ C is the trace product. Linear programming can beviewed as the special case where the given matrices are diagonal. In the literature, it is morecommon to consider semidefinite programs where the matrices are real valued and symmetricrather than Hermitian. However, complex semidefinite programming is easily reduced to thereal case by mapping each Hermitian matrix A to (cid:18) Re A − Im A Im A Re A (cid:19) , see [3].In recent years, many results have been obtained using semidefinite programming, wherethe underlying problem exhibits a large group of symmetries (see [7] for an overview). Thesesymmetries can be exploited to significantly reduce the computational complexity of thesemidefinite program at hand. Indeed, let A be a matrix ∗ -algebra containing the givenmatrices C, A , . . . , A m . For example, A may be the set of matrices invariant under a groupof symmetries of the underlying combinatorial problem, or the matrices C, A , . . . , A m maybelong to the algebra A associated with an association scheme (or coherent configuration).Then the matrix-variable X can be restricted to A without changing the optimum. When anexplicit ∗ -isomorhism φ : A → L i C p i × p i is known, the semidefinite program can be reducedto a semidefinite program in terms of the smaller matrices from L i C p i × p i using the fact thata ∗ -isomorphism preserves positive semidefiniteness. When the algebra A is commutative,for instance when A is an association scheme, the optimization problem is hence reduced toa linear program in a number of variables equal to the dimension of A .When applying block diagonalization to semidefinite programs, it is necessary to have a ∗ -isomorphism available that can be effectively computed. When A consists of the matricesinvariant under the action of a group, the theory of group representations is the tool forderiving an explicit block diagonalization of the invariant algebra A , see [7].In [1, 4, 6] semidefinite programming is used to obtain bounds for error-correcting codes.There the symmetry reduction is essential, reducing the size of the matrices from exponentialto polynomial in the size of the input. The underlying combinatorial problem involves stringsof length n over a finite set of symbols and the main symmetry comes from permuting the n positions in a string. This leads to the consideration of algebras of a specific form.Given a matrix ∗ -algebra B and an integer n , the algebra B ⊗ n admits an action of thesymmetric group S n by defining σ ( B ⊗· · ·⊗ B n ) := B σ − (1) ⊗· · ·⊗ B σ − ( n ) for B , . . . , B n ∈ B , σ ∈ S n and linearly extending this action to the whole of B ⊗ n . The set of symmetric tensorsSym n ( B ) := { A ∈ B ⊗ n | σA = A for all σ ∈ S n } is a matrix ∗ -subalgebra of B ⊗ n .In this paper we derive an explicit block diagonalization of A := Sym n ( B ) in terms of agiven block diagonalization for B . The resulting block diagonalization can be computed inpolynomial time from the block diagonalization of B . When B and its block diagonalizationare defined over the reals (rationals), also the obtained block diagonalization of A is definedover the reals (rationals), up to a scaling by square roots of rationals in the latter case.In Section ?? , we will consider some examples, including the Terwilliger algebra of thebinary- and nonbinary Hamming cube taking respectively B = C × and B = C × ⊕ C × .2 Preliminaries
We denote the natural numbers by N := { , , , . . . } and the positive integers by Z + . For k ∈ Z + , we denote [ k ] := { , . . . , k } . For finite sets S, I and a word a ∈ S I , the weight of a , denoted w ( a ) ∈ N S is given by w ( a ) s := |{ i ∈ I | a i = s }| and counts the number ofoccurences of each element s ∈ S in the word a .Let V be a vector space (say over the complex numbers) with basis v , . . . , v k . A basisof the n -fold tensor product V ⊗ n is given by ( v a ) a ∈ [ k ] n , where v a = v a ⊗ · · · ⊗ v a n . Thesubspace Sym n ( V ) ⊆ V ⊗ n consisting of the symmetric tensors has a basis indexed by alldecompositions µ ∈ N kn of n into k parts: v µ := P a ∈ [ k ] n ,w ( a )= µ v a . In this section we recall some basic facts from the representation theory of finite groups.These results can be found in most textbooks on group representations such as the book byFulton and Harris [2]. Given some finite set S and a finite group G acting on S , we derive a ∗ -algebra isomorphism between the set of complex S × S matrices that are invariant underthe action of G , and a direct sum of full matrix algebras. This isomorphism is expressed interms of representations of G .Let V be a vector space. All vector spaces that we consider will be finite dimensionalover the field of complex numbers. By GL( V ) we denote the set of all invertible lineartransformations of V to itself. Let G be a finite group. Then V is called a G -module if thereis a group homomorphism ρ : G → GL( V ). That is, there is an action of G on V such that g ( c · v + c ′ · w ) = c · g ( v ) + c ′ · g ( w ) for all g ∈ G , c, c ′ ∈ C and v, w ∈ V .A G -homomorphism from V to W , is a linear map φ : V → W that respects the actionof G : φ ( g ( v )) = g ( φ ( v )) for all v ∈ V and g ∈ G . If φ is a bijection, then V and W areisomorphic (as G -modules) and we write V ∼ = W . The set of homomorphisms from V to W is denoted by Hom G ( V, W ) and we denote End G ( V ) := Hom G ( V, V ).Let h· , ·i be a G -invariant inner product on V , that is h x, y i = h g ( x ) , g ( y ) i for all g ∈ G and x, y ∈ V . Such an inner product exists: take h x, y i G := P g ∈ G h g ( x ) , g ( y ) i for any innerproduct h· , ·i . With respect to this inner product, the algebra End G ( V ) becomes a ∗ -algebra,where A ∗ is the adjoint of A ∈ End G ( V ).A module V is called irreducible if V has exactly two submodules: { } and V itself. If W ⊆ V is a submodule, then also W ⊥ := { x ∈ V | h x, y i = 0 for all y ∈ W } is a submodule.Hence it follows that V can be decomposed into pairwise orthogonal irreducible submodules. Theorem 1. (Maschke’s Theorem) Let V be a G -module. Then V has an orthogonal decom-position V = k M λ =1 V λ , V λ = m λ M i =1 V iλ , (3) where V iλ ∼ = V jλ for all i, j = 1 , . . . , m λ and the V λ ( λ = 1 , . . . , k ) are pairwise nonisomorphicirreducible submodules of V . Theorem 2. (Schur’s Lemma) Let V and W be irreducible G -modules. Then either thereis an isomorphism φ : V → W and Hom G ( V, W ) consists of the scalar multiples of φ , or Hom G ( V, W ) consists of the zero map only. Together, Schur’s Lemma and Maschke’s Theorem imply a block diagonalization End G ( V ) ∼ = L kλ =1 C m λ × m λ . To make this explicit, let U , . . . , U k be a complete set of irreducible submod-ules of V , say U λ ∼ = V λ for λ = 1 , . . . , k . Let e λ ∈ U λ \ { } , W λ := { Ae λ | A ∈ End G ( V ) } , andlet B λ be an orthonormal base of W λ . Then we obtain a block diagonalization of End G ( V )as follows. Theorem 3.
The map ψ : End G ( V ) → k M λ =1 C m λ × m λ , (4) A k M λ =1 (cid:0)(cid:10) Ab, b ′ (cid:11)(cid:1) b,b ′ ∈ B λ is a ∗ -algebra isomorphism.Proof. For each λ , the map A ( h Ab, b ′ i ) b,b ′ ∈ B λ is a ∗ -algebra homomorphism since it maps A to its restriction on W λ written as a matrix with respect to an orthonormal base of W λ .Therefore ψ itself is a ∗ -algebra homomorphism.To show injectivity of ψ , suppose that ψ ( A ) = 0. Consider an arbitrary component V iλ in Maschke’s decomposition (3) and let B : U λ → V iλ be a G -isomorphism. Then B can beviewed as an element of End G ( V ) by setting Bx := 0 for all x ∈ U ⊥ λ . Since ψ ( A ) = 0 itfollows that ABe λ = 0 and hence A · V iλ = AB · U λ = AB · C Ge λ = C G · ABe λ = { } . (5)Since this holds for all λ and all i , A is the zero map. Surjectivity follows sincedim End G ( V ) = dim Hom G M λ m λ M i =1 V iλ , M µ m µ M j =1 V jµ (6)= X λ,µ m λ X i =1 m µ X j =1 dim Hom G (cid:0) V iλ , V jµ (cid:1) = X λ m λ , shows that both the dimension of End G ( V ) and the dimension of L λ C m λ × m λ equal P λ m λ .Hence ψ is indeed a bijection.The theorem shows that once a nonzero element e λ in each irreducible submodule (up toequivalence) is identified, the block diagonalization of End G ( V ) can be computed explicitlyonce we can evaluate the inner products h AA ′ e λ , A ′′ e λ i for any e λ and any A, A ′ , A ′′ from asuitable basis of End G ( V ). 4 The algebra
Sym n ( C p × p ) Consider the set [ p ] n of words of length n with symbols in the alphabet [ p ] = { , . . . , p } . Forexample, taking p = 2 gives the set of all binary words of length n written with symbols 1and 2. The symmetric group S n acts on [ p ] n by ( σa ) i := a σ − ( i ) for all a ∈ [ p ] n , σ ∈ S n and i = 1 , . . . , n . This induces a linear action of S n on C [ p ] n by defining ( σx ) a := x σ − a for x ∈ C [ p ] n . Similarly, there is an induced linear action of S n on the set of [ p ] n × [ p ] n matrices M , by defining ( σM ) a,b := M σ − a,σ − b .In this section, we give a block diagonalization of the the matrix ∗ -algebra A = { A ∈ C [ p ] n × [ p ] n | σA = A for all σ ∈ S n } (7)of all S n -invariant matrices in C [ p ] n × [ p ] n . That A is indeed a matrix ∗ algebra follows fromthe facts that S n acts linearly on C [ p ] n × [ p ] n , σA ∗ = ( σA ) ∗ and σ ( AB ) = ( σA )( σB ) for all σ ∈ S n and A, B ∈ C [ p ] n × [ p ] n .For any two words a, b ∈ [ p ] n , define D ( a, b ) ∈ N p × p by( D ( a, b )) i,j := |{ k | a k = i, b k = j }| . (8)Clearly, for a, b, a ′ , b ′ ∈ [ p ] n we have D ( a, b ) = D ( a ′ , b ′ ) if and only if a ′ = π ( a ) , b ′ = π ( b ) forsome π ∈ S n . Let P ( n, p ) := { D ∈ N p × p | T D = n } . (9)For D ∈ P ( n, p ), let A D ∈ A be given by( A D ) a,b := ( D ( a, b ) = D, . (10)Observe that A T D = A D T . Proposition 1.
The matrices A D with D ∈ P ( n, p ) , form a basis for A (as a complex linearspace) and the dimension of A equals (cid:0) n + p − p − (cid:1) .Proof. The matrices A D are nonzero matrices with disjoint support and hence linearly inde-pendent. For any matrix A ∈ A , the value of an entry A a,b only depends on D ( a, b ) and istherefore a linear combination of the matrices A D .Since A is closed under multiplication, there exist numbers c NL,M , L, M, N ∈ P ( n, p ) suchthat A L A M = P N c NL,M A N . Although we will not need the numbers c NL,M , we mention thefollowing fact for completeness.
Proposition 2.
The numbers c NL,M are given by c NL,M = X B p Y r,t =1 (cid:0) N r,t B r, ,t ,...,B r,p,t (cid:1) , (11) where the sum runs over all B ∈ N p × p × p that satisfy P k B r,s,k = L r,s , P k B k,s,t = M s,t and P k B r,k,t = N r,t for all r, s, t ∈ [ p ] . roof. For given words a, c ∈ [ p ] n with D ( a, c ) = N , the number c NL,M equals the numberof words b ∈ [ p ] n with D ( a, b ) = L and D ( b, c ) = M . Fixing the number B r,s,t of positions i = 1 , . . . , n for which a i = r, b i = s, c i = t , the number of feasible b equals 0 unless P k B r,s,k = L r,s , P k B r,k,t = N r,t and P k B k,s,t = M s,t for all r, s, t . In the latter case the solutions b are obtained by partitioning each set S r,t := { i | a i = r, c i = t } into subsets of size B r,s,t for s = 1 , . . . p to choose the entries of b on S r,t . Sym n ( C p × p ) The action of S n on C [ p ] n gives C [ p ] n the structure of an S n -module. Identifying matricesin C [ p ] n × [ p ] n with the corresponding linear maps in GL( C [ p ] n ), the subalgebra A is identifiedwith End G ( C [ p ] n ). Using the representations of the symmetric group, we will find an explicitblock diagonalization of A .A partition λ of n , written λ ⊢ n , is a sequence of nonnegative integers λ ≥ λ ≥ · · · ≥ λ n with n = λ + λ + · · · + λ n . Partition λ is said to have k parts if exactly k of the numbers λ i are nonzero. A Ferrers diagram of shape λ is an array of n boxes, taking the first λ i boxesfrom the i -th row of an n × n matrix of boxes. The j -th box in row i is referred to as the boxin position ( i, j ). We also number the boxes from 1 to n according to the lexicographic orderon their positions: the box in position ( i, j ) has number λ + · · · + λ i − + j . The dual partition λ ∗ of n gives the column lengths of a Ferrers diagram of shape λ : λ ∗ i = |{ j | λ j ≥ i }| .A tableau t of shape λ is a filling of the boxes in the Ferrers diagram with integers.Here all entries will be from the set [ p ]. The number in box k is denoted by t ( k ). Fixing apartition λ ⊢ n , each element a of [ p ] n is identified with the tableau t given by t ( k ) := a k .For an element σ ∈ S n and a tableau t , we define ( σt )( k ) := t ( σ − ( k )). A tableau t is called semistandard if the entries are increasing in each column and non-decreasing in each row.We denote the set of semistandard tableaux of shape λ with entries from [ p ] by T λ,p .Given a partition λ ⊢ n , we can associate with λ two subgroups of S n . The group C λ consists of the permutations of boxes within the columns of the diagram, and the group R λ consists of the permutations that permute the boxes within the rows. Two λ -tableaux t, t ′ are called (row-)equivalent , written t ∼ t ′ , when t ′ = πt for some π ∈ R λ .Recall that the complex vector space V = C [ p ] n is an S n -module. We denote by χ a ∈ V the standard basis vector corresponding to the word a . Then the action of S n on V is givenby σ ( X a ∈ [ p ] n c a χ a ) := X a ∈ [ p ] n c a χ σa (12)for all σ ∈ S n and c a ∈ C for all a ∈ [ p ] n .Let λ be a partition of n into at most p parts. Define the tableau t λ by filling the positionsin the i -th row with i ’s. Given any tableau t of shape λ , we define e t ∈ V by e t := X σ ∈ C λ sgn( σ ) X t ′ ∼ t χ σ ( t ′ ) . (13)Define S λ := C S n · e λ , where e λ := e t λ = P σ ∈ C λ sgn( σ ) χ σ ( t λ ) .6 heorem 4. The S λ , where λ runs over all partitions of n into at most p parts, form acomplete set of pairwise non-isomorphic, irreducible submodules of V . The irreducible modules S λ are called Specht modules . The proof is standard and can befound for example in [5].Let D ∈ P ( n, p ) and let λ = D T . If λ is nonincreasing, we view it as a partition of n andmake a tableau t ( D ) of shape λ and weight µ = D as follows: the i -th row of t ( D ) contains D j,i symbols j , and we make the rows of t ( D ) non-decreasing. Observe that D ( t ( D ) , t λ ) = D .We have the following lemma. Lemma 1.
Let A = A D ∈ A and let λ ′ := D T . Then Ae λ = ( e t ( D ) if λ ′ = λ otherwise. (14) Proof.
Recall that we identify [ p ] n with the set of all tableau of shape λ and entries from[ p ]. For any such tableau t , we have Aχ t = P s | D ( s,t )= D χ s . Since D ( s, t ) T = w ( t ), this sumequals zero when w ( t ) = λ ′ . In particular, Aχ t λ = ( λ = λ ′ , P s ∼ t ( D ) χ s if λ = λ ′ . (15)The lemma now follows by writing out the definition of Ae λ : Ae λ = X σ ∈ C λ sgn( σ ) Aχ σ ( t λ ) (16)= X σ ∈ C λ sgn( σ ) σ ( Aχ t λ ) . The lemma implies that the linear space A e λ is spanned by the vectors e t (where t ∈ [ p ] n is a tableau of shape λ ). The following theorem selects a subset of the tableaux to obtain abasis. Theorem 5.
The e t , t ∈ T λ,p constitute a basis of span { e t | t ∈ [ p ] n is a tableau of shape λ } . The proof is standard and can be found for example in [5].In general the basis { e t } ( t ∈ T λ,p ) of A e λ , is not orthonormal. However, let B := ( e t ) t bethe matrix with the e t as columns and let G λ := B T B be the Gram matrix of the e t . Takea Cholesky decomposition ( G λ ) − =: R λ R T λ , then the columns of BR λ form an orthonormalbase of A e λ . Theorem 6.
The map ψ : A → M λ C m λ × m λ (17) A M λ R T λ ( h Ae s , e t i ) s,t R λ is a ∗ -isomorphism. roof. This follows directly from Theorem 3 since the e t , t ∈ T λ,p form a basis of the space A e λ , where the e λ generate the irreducible submodules of V . Remark 1.
Although the vectors e t have length exponential in n , we can compute their innerproducts efficiently (see next section). This implies that we can find the Gram matrices G λ efficiently. However, in applications to semidefinite programming, we only need ψ to preservepositive semidefiniteness. Hence we can neglect the matrices R λ and obtain a bijection ψ ′ : A → M λ C m λ × m λ (18) A M λ ( h Ae s , e t i ) s,t that preserves positive semidefiniteness. ψ In this section, we wil show how the map ψ can be computed efficiently. That is, given apartition λ of n into at most p parts, and given semistandard tableaux s, t ∈ T λ,p of shape λ ,we compute for every D ∈ P ( n, p ), the inner product h A D e s , e t i .For A ∈ C [ p ] × [ p ] n the linear map A → C given by A D
7→ h A D , A i is conveniently expressedusing polynomials in the p variables x i,j , i, j = 1 , . . . , p : f A := X D ∈ P ( n,p ) x D h A D , A i , (19)where the shorthand notation x D := Q pi,j =1 x D i,j i,j is used.Define for k = 0 , , . . . , p the polynomials Q k by setting Q ( x ) := 1 and Q k ( x ) := k ! det x , · · · x ,k ... . . . ... x k, · · · x k,k (20)for k = 1 , , . . . , p . The polynomials Q k have at most p ! terms and can, for fixed p , be com-puted in constant time. Given a partition λ ⊢ n into at most p parts, define the polynomial P λ by P λ := n Y i =1 Q λ ∗ i = Q λ − λ · · · Q λ p − − λ p p − · Q λ p p . (21)For fixed p , computing P λ can be done in time O ( n dim( A )). Proposition 3.
Let λ ⊢ n be a partition with at most p parts. Then P λ = f e λ e T λ . (22)8 roof. Writing out the definition of A D and e λ we obtain X D ∈ P ( n,p ) x D · D A D , e λ e T λ E = X D ∈ P ( n,p ) x D X ρ,τ ∈ C λ sgn( ρ )sgn( τ )( A D ) ρt λ ,τt λ (23)= X D ∈ P ( n,p ) x D · | C λ | X σ ∈ C λ sgn( σ )( A D ) σt λ ,t λ = | C λ | X σ ∈ C λ sgn( σ ) x D ( σt λ ,t λ ) . Here we used the substitution σ := τ − ρ so thatsgn( ρ )sgn( τ )( A D ) ρt λ ,τt λ = sgn( σ )( A D ) σt λ ,t λ . (24)Any σ ∈ C λ corresponds to an n -tuple ( σ , . . . , σ n ), where σ i ∈ S λ ∗ i is the restriction of σ to the i -th column. Observe that ( t λ ) i,j = i and ( σt λ ) i,j = σ − j ( i ) and hence x D ( σt λ ,t λ ) = n Y j =1 λ ∗ j Y i =1 x σ − j ( i ) ,i . (25)Using this, we obtain: | C λ | X σ ∈ C λ sgn( σ ) x D ( σt λ ,t λ ) = | C λ | n Y j =1 ( X σ j ∈ S λ ∗ j sgn( σ j ) λ ∗ j Y i =1 x σ − j ( i ) ,i ) (26)= | C λ | n Y j =1 λ ∗ j ! Q λ ∗ j = P λ . Since we can perform multiplication in A in polynomial time, we now have a polynomialtime algorithm to compute h Ae t , e t ′ i = D A T D ( t ′ ,t λ ) A ( D ) A D ( t,t λ ) , e λ e T λ E for any t, t ′ ∈ T λ,p and D ∈ P ( n, p ), hence obtaining the map ψ explicitly.Below we show how we can speed up the computation, by avoiding the computationallycostly multiplication in A .For i, j ∈ [ p ], let A i → j ∈ A be defined by( A i → j ) a,b := ( k such that a k = i, b k = j and a l = b l for all l = k ,0 otherwise . (27)For i, j ∈ [ p ], let E i,j be the [ p ] × [ p ] matrix with ( E i,j ) i,j = 1 and all other entries equalto 0. 9 roposition 4. For any D ∈ P ( n, p ) we have: A D A i → j = X k | A k,i > ( D k,j + 1) A D − E k,i + E k,j , (28) A i → j A D = X k | A j,k > ( D i,k + 1) A D − E j,k + E i,k . Proof.
To prove he first line, consider two words a, b ∈ [ p ] n . The entry ( A D A i → j ) a,b equals thenumber of c ∈ [ p ] n such that D ( a, c ) = D and b is obtained from c by changing an i into a j insome position h . If a h = k , then D ( a, b ) = D − E k,i + E k,j . When D ( a, b ) = D − E k,i + E k,j ,the number of possible c equals D ( a, b ) k,j = D k,j + 1.For the second line, observe that A i → j A D = ( A D T A j → i ) T .Given a decomposition µ = ( µ , . . . , µ p ) of n ( µ i ≥ I µ := A µ E , + ··· + µ p E p,p . Proposition 5. If D ∈ P ( n, p ) is lower triangular with µ := D T , then Y i>j D i,j ! A D = ( p − Y j =1 p Y i = j +1 A D i,j i → j ) I µ . (29) Proof.
For given i = j and D ∈ P ( n, p ) with D j,j ≥ D j,k = 0 for all k = j , Proposition(4) gives A i → j A D = ( D i,j + 1) A D − E j,j + E i,j . (30)Let ( i , j ) , . . . , ( i m , j m ) be m pairs of indices with i k = j k for all k and i k > j l whenever k < l ,and let s , . . . , s m be nonnegative integers. Let D := µ E , + · · · + µ p E p,p + P mk =1 s i ( E i k ,j k − E j k ,j k ) and suppose that D is nonnegative. Then from (30) we obtain A s m i m → j m · · · A s i → j I µ = s m ! · · · s ! A D , (31)by induction on m . This implies the statement in the proposition.Let us denote d i → j := p X s =1 x i,s ∂∂x j,s , d ∗ i → j := p X s =1 x s,j ∂∂x s,i . (32)In terms of polynomials, Proposition (4) now gives Proposition 6.
For i = j and a matrix A , we have f A i → j A = d i → j f A (33) f AA i → j = d ∗ i → j f A . roof. We have: f AA i → j = X D x D h A D A j → i , A i = X D x D X k | D k,i > ( D k,i + 1) (cid:10) A D − E k,j + E k,i , A (cid:11) (34)= X k | D k,i > X D x D ( D k,i + 1) (cid:10) A D − E k,j + E k,i , A (cid:11) = X k | D k,i > X D ′ x D ′ + E k,j − E k,i D ′ k,i h A D ′ , A i = p X k =1 x k,j ∂∂x k,i f A . Here we used the substitution D ′ := D − E k,i + E k,j . The proof for f A i → j A is similar. Theorem 7.
Let t ′ , t ′′ be semistandard λ -tableau and let D ′ := D ( t ′ , t λ ) , D ′′ := D ( t ′′ , t λ ) .Then X D ∈ P ( n,p ) x D h A D e t ′ , e t ′′ i = q · Y i>j ( D ′ i,j ! D ′′ i,j !) − , (35) where q := p − Y j =1 p Y i = j +1 (( d i → j ) D ′′ i,j ( d ∗ j → i ) D ′ i,j ) ◦ P λ . (36) Proof.
We want to compute X D x D h A D e t ′ , e t ′′ i = X D x D D A D , A D ′′ e λ e T λ A T D ′ E = f A D ′′ E λ A T D ′ . (37)By Proposition (5) we obtain Y i>j ( D ′ i,j ! D ′′ i,j !) · A D ′′ E λ A T D ′ = p − Y j =1 p Y i = j +1 A D ′′ i,j i → j I λ E λ I λ ( p − Y j =1 p Y i = j +1 A D ′ i,j i → j ) T (38)= p − Y j =1 p Y i = j +1 A D ′′ i,j i → j E λ ( p − Y j =1 p Y i = j +1 A D ′ i,j i → j ) T Using Proposition (6) we derive Y i>j ( D ′ i,j ! D ′′ i,j !) · f A D ′′ E λ A T D ′ = p − Y j =1 p Y i = j +1 (( d i → j ) D ′′ i,j ( d ∗ j → i ) D ′ i,j ∗ ) ◦ P λ . (39)11 The general case
Let B be a complex matrix algebra with basis R , . . . , R s , and let A := Sym n ( B ). Recall thatwe have a basis { R µ | µ ∈ N sn } given by R µ := P x ∈ [ s ] n | w ( x )= µ ⊗ ni =1 R x i .Assume that a block diagonalization φ : B → t M i =1 C p i × p i , B t M i =1 φ i ( B ) (40)is given. In the following, we will describe a block diagonalization of A in terms of φ andwith respect to the basis { R µ } .First observe that the isomorphism φ ⊗ n : B ⊗ n → ( ⊕ ti =1 C p i × p i ) ⊗ n gives an isomorphism A →
Sym n ( ⊕ ti =1 C p i × p i ) by restriction to A . Now observe, that by removing multiple copiesof identical blocks in Sym n ( ⊕ ti =1 C p i × p i ), we have an isomorphismSym n ( ⊕ ti =1 C p i × p i ) → ⊕ µ ∈ N tn ⊗ Sym µ i ( C p i × p i ) (41)Indeed, let { E ( k,l ) | k, l = 1 , . . . , m } denote the standard basis of C m . This gives a basis { E D | D ∈ N m × mn } given by E D := P x ∈ ([ m ] × [ m ]) n ,w ( x )= D E x . Similarly, the standard basis { E ( i,k,l | i = 1 , . . . , t, k, l = 1 , . . . , p i } of ⊕ ti =1 C p i × p i gives a basis { E D | D = D ⊕ · · · ⊕ D t } of Sym n ( ⊕ ti =1 C p i × p i ) where E D := P x ∈ ([ p ] × [ p ] ∪···∪ [ p t ] × [ p t ]) n | w ( x )= D ⊕···⊕ D t . Then theisomorphism is given by A D
7→ ⊕ µ δ T D ,µ · · · δ T D t ,µ t ⊗ ti =1 A D i . (42)This yields an isomorphism ψ : A → M µ ∈ N tn t O i =1 Sym µ i ( C p i × p i ) . (43)To make this map explicit in terms of the given bases, we use the following lemma. Lemma 2.
Let V be a vector space with bases { u , . . . , u m } and { v , . . . , v m } that are relatedby v i = P j c j,i u j . Introducing indeterminates x , . . . , x m , the bases ( u µ ) µ ∈ N mn and ( v ν ) ν ∈ N mn of Sym n ( V ) are related by v ν = X µ u µ · ( x c , + · · · + x m c ,m ) µ · · · ( x c m, + · · · + x m c m,m ) µ t [ x ν ] . (44)12 roof. We compute P ν ∈ N mn x ν v ν . X ν ∈ N mn x ν v ν = X y ∈{ ,...,m } n ⊗ ni =1 x y i v y i = X y ∈{ ,...,m } n ⊗ ni =1 x y i m X j =1 c j,i u j = X y,z ∈{ ,...,m } n ⊗ ni =1 x y i c z i ,y i u z i = X z ∈{ ,...,m } n ⊗ ni =1 u z i m X j =1 x j c z i ,j = X µ u µ m X j =1 x j c ,j µ · · · m X j =1 x j c m,j µ m . (45)Application of the lemma yields the following result. Theorem 8.
The map ψ is given by ψ ( X ν y ν A ν ) = M µ X D =( D ,...,D t ) y D t O i =1 A D i , (46) where T D i = µ i . The coefficients y D are given by y D = X ν y ν t Y j =1 p j Y k,l =1 s X i =1 x i ( φ j ( R i )) k,l ! ( D j ) k,l [ x ν ] . (47) Proof.
Expressing the basis { ψ ( A ν ) | ν ∈ N sn } in terms of the basis { A D } using Lemma 2yields the claimed result.Using the block diagonalization for each of the algebras Sym µ i ( C p i × p i ) as described insection 3.1, we obtain a block diagonalization A → M µ M λ ,...,λ t t O i =1 C T λi,pi ×T λi,pi (48)of A . 13 Examples from coding theory
Terwilliger algebra of the binary Hamming scheme
The algebra A := Sym n ( C × ) is referred to in the literature as the Terwilliger algebra of theHamming scheme. The matrices D ∈ P ( n,
2) are usually indexed by 3 parameters i, j, t bysetting D ti,j := (cid:18) n + t − i − j i − tj − t t (cid:19) , A ti,j := A D ti,j . (49)Here 0 ≤ t ≤ i, j and i + j − t ≤ n . In [6] an explicit block diagonalization was given andused to compute semidefinite programming bounds for binary codes. Here we show that ourmethod gives the same block diagonalization.Since p = 2, the partitions λ = ( n − k, k ) are indexed by k = 0 , . . . , ⌊ n ⌋ . Give k , thesemistandard tableaux of shape ( n − k, k ) are indexed by i = k, . . . , n − k by placing a 2 in i − k of the last n − k boxes of the first row and in all k boxes of the second row of the tableau.Lets denote this tableau by t k,i and denote e t k,i by e k,i . We have D ( t k,i , t k,k ) = (cid:0) n − ii − k k (cid:1) .Hence for give i, j : X D x D h A D e k,i , e k,j i = 2 k ( i − k )!( j − k )! d j − k → ( d ∗ → ) i − k ( x , x , − x , x , ) k x n − k , . (50)It is easy to see that d → ( x , x , − x , x , ) = d ∗ → ( x , x , − x , x , ) = 0. Hence weobtain: X D x D h A D e k,i , e k,j i = 2 k ( i − k )!( j − k )! ( x , x , − x , x , ) k d j − k → ( d ∗ → ) i − k x n − k , (51)= 2 k (cid:0) n − ki − k (cid:1) ( j − k )! ( x , x , − x , x , ) k d j − k → x i − k , x n − k − i , = 2 k (cid:0) n − ki − k (cid:1) ( x , x , − x , x , ) k j − k X s =0 (cid:0) n − k − is (cid:1)(cid:0) i − kj − k − s (cid:1) . In this sum, only monomials x D with D = D ti,j for some t , occur and they have coefficient h A D e k,i , e k,j i k (cid:0) n − ki − k (cid:1) j − k X s =0 (cid:0) n − k − is (cid:1)(cid:0) i − kj − k − s (cid:1)(cid:0) kj − t − s (cid:1) ( − j − t − s . (52)Note that for i = j the vectors e k,i and e k,j are orthogonal as they have disjoint support.Taking t = i = j we see that h e k,i , e k,i i = 2 k (cid:0) n − ki − k (cid:1) . (53)It follows that the block diagonalization is given by: ψ : A ti,j ⌊ n ⌋ M k =0 (cid:16) δ i ′ ,i δ j ′ ,j (cid:0) n − ki − k (cid:1) − / (cid:0) n − kj − k (cid:1) − / β ti,j,k (cid:17) n − ki ′ ,j ′ = k , (54)14here β ti,j,k := ( − j − t − s (cid:0) n − ki − k (cid:1) j − k X s =0 (cid:0) n − k − is (cid:1)(cid:0) i − kj − k − s (cid:1)(cid:0) kj − t − s (cid:1) . (55) Remark 2.
This is the same block diagonalization as was given by Schrijver[6], except thatthere a different expression (but the same value!) was used for β ti,j,k , namely β ti,j,k = n X u =0 ( − u − t (cid:0) ut (cid:1)(cid:0) n − ku − k (cid:1)(cid:0) n − k − ui − u (cid:1)(cid:0) n − k − uj − u (cid:1) . (56) More algebras for binary codes
Fix a positive integer t and let X := { , } t . The symmetric group of two elements, S ,acts on X by exchanging the symbols 0 and 1 (in all t positions). This induces an actionof S on the set of X × X matrices by simultaneous permutation of the rows and columns.Let B := { A ∈ C X × X | A is S -invariant } be the algebra of matrices invariant under thisaction. The algebra A n := Sym n ( B ) can be indentified with the set of matrices with rowsand columns indexed by all t × n binary matrices I that are invariant under all permutationsof the indices induced by either permuting the n rows of I or by action of S on any subsetof the columns of I .Since B ∼ = C { , } t − ×{ , } t − ⊕ C { , } t − ×{ , } t − by sending (cid:18) A BB A (cid:19) → A + B ⊕ A − B ,Theorem... gives an explicit block diagonalization of A n . The simplest case, t = 1 gives adiagonalization of the Bose-Mesner algebra of the Hamming cube. In the case t = 2 we seethat A n ∼ = ⊕ ni =0 Sym ⊗ ( C × , i )Sym n − i ( C × ), a direct sum of tensor products of Terwilligeralgebras for the Hamming scheme. Terwilliger algebra of the nonbinary Hamming scheme
Let q ≥ B ⊂ C q × q be the set of matrices with rows and columnsindexed by { , , . . . , q − } that are invariant under simultaneous permutation of the rowsand columns in { , . . . , q − } . The algebra B is easily seen to have dimension 5, where B , . . . , B form a basis by defining X i =1 x i B i := x x ··· ··· x x x x ··· x ... x ... ... ...... ... ... ... x x x ··· x x . (57)A block diagonalization B → C × ⊕ C × is given by φ ( x B + · · · + x B ) = (cid:16) x x √ q − x √ q − x + x ( q − (cid:17) ⊕ ( x − x ) (58)for x , . . . , x ∈ C . 15he algebra A q,n := Sym n ( B ) is known as the Terwilliger algebra of the nonbinary Ham-ming scheme. It can be used for deriving bounds on the size of nonbinary codes from semidef-inite programming, see [1].Applying Theorem 8, we obtain a block diagonalization of A q,n given by ψ ( A ν ) = n M w =0 X D = “ a bc n − w − a − b − c ” ⊕ ( w ) y D A “ a bc n − w − a − b − c ” , (59) y D = x a x b x c ( x + ( q − x ) n − w − a − b − c ( x − x ) w ( q − b + c [ x ν ] . (60)Restricting to D with y D = 0 we obtain: ψ ( A ν ) = n M w =0 A ( ν ν ν ν + ν − w ) X g (cid:0) ν + ν − wg (cid:1)(cid:0) wν − g (cid:1) ( q − g ( − ν − g ( q − ν ν . (61)Using the block diagonalization of Sym n − w ( C × ), we obtain a block diagonalization of A q,n . This agrees with the block diagonalization found in [1]. Acknowledgments
I want to thank Jan Draisma, Lex Schrijver and Frank Vallentin for helpful comments anddiscussions.
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