BOSE-MESNER ALGEBRAS ATTACHED TO INVERTIBLE JONES PAIRS 4 0 0 2 ADACHANANDCHRISGODSIL n a Abstract. In1989,VaughanJonesintroducedspinmodelsandshowedthat J theycouldbeusedtoformlinkinvariantsintwodifferentways—byconstruct- 6 ingrepresentationsofthebraidgroup,orbyconstructingpartitionfunctions. 2 These spin models were subsequently generalized to so-called 4-weight spin models by Bannai and Bannai; these could be used to construct partition ] functions,butdidnotleadtobraidgrouprepresentationsinanyobviousway. O Jaegershowedthatspinmodelswereintimatelyrelatedtocertainassociation schemes. Yamadagaveaconstructionofasymmetricspinmodelon4nvertices C fromeach4-weightspinmodelonnvertices. h. In this paper we build on recent work with Munemasa to give a different t proof to Yamada’s result, and we analyse the structure of the association a schemeattached tothisspinmodel. m [ 2 v 0 7 1. Introduction 3 Spin models are a special class of matrices introduced by Jones in [8] as a tool 3 0 forcreatinglinkinvariants. Therearetwostrandstotheirsubsequentdevelopment 3 that are of interest to us. First, Jaeger and Nomura showed that all spin models 0 could be realized as matrices in association schemes (see [10]). Hence spin models / h haveacombinatorialaspectand,perhapsmoreimportantly,thesearchfornewspin t models was reducedto the searchfor certainspecialclasses ofassociationschemes. a m (This means that the search space is discrete rather than continuous.) Thesecondstrandwasthedevelopmentofmoregeneralclassesofmodels,culmi- : v nating in the four-weightspin models of Bannai and Bannai [2]. These models are i formed from a pair of matrices; they still provided link invariants, but apparently X lacked the intimate connection to association schemes. r a In [4], Munemasa and the present authors developed a new approach to spin models, based on what we called Jones pairs. We showed that these included the four-weight spin models as a special case. As a result we were able to show that each four-weight spin model determines a pair of association schemes. In [11], Yamada showed that each four-weight spin model of order n embeds in a very natural way in a spin model of order 4n. We give a complete and different proofto Yamada’sresult. Inaddition,the toolswedevelopin Sections2to 5 allow us to analyze the structure of N , which was not investigated in [11]. V 1991 Mathematics Subject Classification. Primary05E30;Secondary 20F36. SupportfromaNationalSciencesandEngineeringCouncilofCanadaoperatinggrantisgrate- fullyacknowledged bythesecondauthor. 1 2 ADACHANANDCHRISGODSIL 2. Invertible Jones Pairs Given two matrices A and B of the same order, we use A◦B to denote their Schur product, which has (A◦B) =A B . i,j i,j i,j If all entries of A are non-zero, then we say A is Schur invertible and define its Schur-inverse,A(−), by A(−) = 1 . i,j A i,j Equivalently, we have A(−)◦A=J, where J is the matrix of all ones. For any n×n matrix C, we define two linear operators X and ∆ as follows: C C X (M):=CM, ∆ (M):=C◦M for all M ∈M (C). C C n Given a linear operator Y on M (C), we use YT to denote its adjoint relative to n the non-degeneratebilinear form tr(MTN) on M (C), and call it the transpose of n Y. It is easy to see that XT =X , ∆T =∆ . C CT C C A Jones pair is a pair of n×n complex matrices (A,B) such that X and ∆ are A B invertible and X ∆ X = ∆ X ∆ , (2.1) A B A B A B X ∆ X = ∆ X ∆ . (2.2) A BT A BT A BT NotethatX and∆ areinvertibleonlyifAisinvertibleandB isSchurinvertible. A B Itisalsoeasytoobservethat(A,B)isaJonespairifandonlyif(A,BT)isaJones pair. Jones pairs are designed to give representation of braid groups using Jones’ construction. Please see Section 2 of [4] for a description of the construction. An n×n matrix W is a type-II matrix if WW(−)T =nI. Note that a type-II matrix is invertible with respect to both matrix multiplication and the Schur product. We say that a Jones pair (A,B) is invertible if A is Schur invertible and B is invertible. Theorems 7.1 and 7.2 of [4] imply that a Jones pair (A,B) is invertible if and only if A and B are type-II matrices. Let W , W , W and W be n× n complex matrices and let d be such that 1 2 3 4 d2 =n. A four-weight spin model is a 5-tuple (W ,W ,W ,W ;d) that satisfies 1 2 3 4 W =W (−)T, W =W (−)T, (2.3) 3 1 2 4 W W =nI, W W =nI, (2.4) 1 3 2 4 n (W ) (W ) (W ) = d(W ) (W ) (W ) , (2.5) 1 k,h 1 h,i 4 h,j 4 i,j 1 k,i 4 k,j h=1 X n (W ) (W ) (W ) = d(W ) (W ) (W ) . (2.6) 1 h,k 1 i,h 4 j,h 4 j,i 1 i,k 4 j,k h=1 X From (2.3) and (2.4), we see that both W and W are type-II matrices and they 1 4 determine W and W , respectively. Furthermore, it is straightforward to verify 3 2 that Equations (2.5) and (2.6) are equivalent to Equations (2.1) and (2.2) when W =dA and W =B. 1 4 BOSE-MESNER ALGEBRAS ATTACHED TO INVERTIBLE JONES PAIRS 3 Jaeger showed in [6] that (A,B) and (C,B) are invertible Jones pairs if and only if C =DAD−1 for some invertible diagonal matrix D. We say that these two invertible Jones pairs areodd-gauge equivalent. Proposition7 of [6] states that for everyinvertibleJonespair(A,B),thereexistsaninvertiblediagonalmatrixD such that DAD−1 is symmetric. Since odd-gauge equivalent invertible Jones pairs give thesamelinkinvariants,wesuffernolossbyconsideringonlyinvertibleJonespairs whose first matrix is symmetric. 3. Nomura Algebras We startthis section by defining the Nomura algebras N and N′ of a pair A,B A,B of n×n matrices. When A is a type-II matrix and B = A(−), our construction gives the Nomura algebras discussed in [7] and [10]. The definitions here are taken from [4]. Let A and B be n×n matrices, let e ,...,e be the standard basis vectors in 1 n Cn and form the n2 column vectors Ae ◦Be for i,j =1,...,n. i j We define N to be the set of matrices of which Ae ◦Be is an eigenvector, for A,B i j all i,j = 1,...,n. This set of matrices is closed under matrix multiplication and contains the identity matrix I . n For each matrix M ∈ N , we use Θ (M) to denote the n×n matrix that A,B A,B satisfies M(Ae ◦Be )=Θ (M) (Ae ◦Be ). i j A,B i,j i j We view Θ as a linear map from N to M (C) and we use N′ to denote A,B A,B n A,B the image of N . By the definition of Θ , we have A,B A,B Θ (MN)=Θ (M)◦Θ (N). A,B A,B A,B Consequently the space N′ is closed under the Schur product. Since I ∈N , A,B n A,B the matrix Θ (I ) = J belongs to N′ . We conclude that N′ is a commu- A,B n n A,B A,B tative algebra with respect to the Schur product. If A is invertible, then the columns of A are linearly independent. Further if B is Schur invertible, then for any j {Ae ◦Be ,Ae ◦Be ,...,Ae ◦Be } 1 j 2 j n j is a basis of Cn. In this case, the map Θ is an isomorphism from N , as A,B A,B an algebra with respect to the matrix multiplication, to N′ , as an algebra with A,B respect to the Schur product. We conclude from the commutativity of N′ that A,B N is commutative with respect to matrix multiplication. A,B The following result is called the Exchange Lemma. It will serve as a powerful toolinSections6and7. TheproofofTheorem3.2alsodemonstratestheusefulness of this lemma. 3.1. Lemma ([4], Lemma 5.1). [Exchange] If A,B,C,Q,R,S ∈M (C) then n X ∆ X =∆ X ∆ A B C Q R S if and only if X ∆ X =∆ X ∆ . A C B R Q ST 3.2. Theorem. If A and B are n × n type-II matrices, then the following are equivalent: 4 ADACHANANDCHRISGODSIL (a) R∈N and S =Θ (R). A,B A,B (b) X ∆ X =∆ X ∆ . R B A B A S (c) X ∆ X =∆ X ∆ . R A B A B ST (d) ∆BTXB(−)T∆nR =XST∆A(−)TXAT. (e) ∆A(−)TXAT∆nRT =XS∆BTXB(−)T. Proof. The equivalence of (a) and (b) follows from Theorem 6.2 of [4]. Applying the Exchange Lemma to (b) gives (c), which is equivalent to ∆A(−)XR∆A =XB∆STXB−1. (3.1) Applying the Exchange Lemma to Equation (3.1) again, we get ∆RXA(−)∆AT =XB∆B−1XST. Now we have B−1 =n−1B(−)T and A(−) =n(A−1)T because A and B are type-II matrices. The above equation becomes ∆RXn(A−1)T∆AT =XB∆n−1B(−)TXST, which leads to ∆BTXB−1∆nR =XST∆A(−)TXn−1AT. (3.2) We get(d)after multiplying bothsides of Equation(3.2) byn andreplacingnB−1 by B(−)T. Taking the transpose of both sides of Equation (3.2) gives ∆nRX(B−1)T∆BT =Xn−1A∆A(−)TXS and ∆ATXA(−)T∆nR =XS∆B(−)TXBT. We get (e) after applying the Exchange Lemma to the above equation. Now we state an easy consequence of Theorem 3.2 (b). 3.3. Corollary ([4], Lemma 10.2). Let A and B be n × n type-II matrices. If R∈N then A,B RT ∈NA(−),B(−), and ΘA(−),B(−)(RT)=ΘA,B(R). 4. Nomura Algebras of a Type-II Matrix When A is a type-II matrix and B = A(−), existing papers such as [7] use N , N′ and Θ to denote N , N′ and Θ , respectively. The algebra N A A A A,B A,B A,B A is called the Nomura algebra of A. We now present some results on N due to A Jaeger,Matsumoto and Nomura [7] which we will use later. When B =A(−), Condition 3.2 (e) becomes ∆A(−)TXAT∆nRT =XS∆A(−)TXAT, and it implies Θ (S)=Θ (Θ (R))=nRT. (4.1) AT AT A We conclude that if R∈N then Θ (R)∈N and RT ∈N′ . Hence A A AT AT N′ ⊆N and dim(N )=dim(NT)≤dim(N′ ). A AT A A AT Similarly AT is also a type-II matrix, so N′ ⊆N and dim(N )≤dim(N′). AT A AT A BOSE-MESNER ALGEBRAS ATTACHED TO INVERTIBLE JONES PAIRS 5 Therefore N′ =N and N′ =N , which implies that N and N are closed A AT AT A A AT underbothmatrixmultiplicationandtheSchurproduct. ItalsoimpliesthatN = A N′ isclosedunderthetranspose. SinceAisinvertibleandA(−)isSchurinvertible, AT the map Θ is an isomorphism from N to N′. Hence N is commutative with A A A A respecttomatrixmultiplication. Insummary,thealgebraN iscommutativewith A respect to matrix multiplication, is also closed under the transpose and the Schur product, and contains I and J. In other words, N is a Bose-Mesner algebra. A We now investigate the properties of the map Θ . Let M and N be matrices in A N . Since Θ : N → N is an isomorphism, there exist M′ and N′ in N A AT AT A AT such that Θ (M′)=M and Θ (N′)=N. Hence AT AT Θ (M ◦N) = Θ (Θ (M′)◦Θ (N′)) A A AT AT = Θ (Θ (M′N′)) A AT which equals n(M′N′)T by Equation (4.1). Since Θ (M)=Θ (Θ (M′))=nM′T A A AT and Θ (N)=nN′T, we have A 1 Θ (M ◦N) = (nN′T)(nM′T) A n 1 = Θ (N)Θ (M) A A n 1 = Θ (M)Θ (N), A A n the last equality results from the commutativity of N′. Now we conclude that Θ A A swaps matrix multiplication with the Schur product. Furthermore, applying 1Θ to the two rightmost terms of Equation (4.1) gives n A 1 Θ (Θ (Θ (R)))=Θ (RT). n A AT A A It follows from Equation (4.1) that the left-hand side equals Θ (R)T. Thus Θ A A and the transpose commute. From Corollary 3.3, we see that ΘA(−)(R)=ΘA(R)T. Also note that by Equation (4.1), we have Θ (J)=Θ (Θ (I))=nI. A A AT We call Θ a duality map from N to N and say that these two Bose-Mesner A A AT algebrasform a formally dual pair. If N =N and Θ =Θ , we say that it is A AT A AT formally self-dual. A spin model is an n×n matrix W such that (W,W,W(−),W(−);d) is a four- weightspinmodel,ford2 =n. ItfollowsfromSection9of[4]thatW isaspinmodel if and only if (d−1W,W(−)) is an invertible Jones pair. In [7], Jaeger, Matsumoto andNomuragavethefollowingcharacterizationofaspinmodelW usingitsNomura algebra N . W 4.1. Theorem ([7], Theorem 11). Suppose W is a type-II matrix. Then W ∈N W if and only if cW is a spin model for some non-zero scalar c. In this case, N =N W WT is a formally self-dual Bose-Mesner algebra with duality map Θ =Θ . W WT 6 ADACHANANDCHRISGODSIL 5. Nomura Algebras of an Invertible Jones Pair WestudytherelationamongthedifferentNomuraalgebrasofaninvertibleJones pair. 5.1. Theorem ([1], Theorem 3). If (A,B) is an invertible Jones pair, then N =N =N =N , A AT B BT the duality maps satisfy Θ =Θ and Θ =Θ . A AT B BT Bannai,GuoandHuang[1]provedthisresultforfour-weightspinmodels,which are equivalent to invertible Jones pairs. For an alternate proof using the Nomura algebras of A and B, see Section 10 of [4]. Let A and B be type-II matrices. We see from Theorem 3.2 (a) and (b) that (A,B) is an invertible Jones pair if and only if A ∈ N ∩N , Θ (A) = B A,B A,BT A,B and Θ (A) = BT. The next two results provide some insights to the relations A,BT among N , N′ and N . A,B A,B A 5.2. Theorem ([4], Theorem 10.3). Let A and B be n×n type-II matrices. If F ∈N , G∈N and H ∈N , then F ◦G, and G◦H belong to N and A A,B B A,B Θ (F ◦G) = n−1Θ (F) Θ (G), A,B A A,B Θ (G◦H) = n−1Θ (G) Θ (H)T. A,B A,B B 5.3. Theorem ([4], Theorem 10.4). Let A and B be n×n type-II matrices. If F,G∈N , then F ◦GT ∈N ∩N and A,B A B Θ (F ◦GT) = n−1Θ (F) Θ (G)T, A A,B A,B Θ (F ◦GT) = n−1Θ (F)T Θ (G). (5.1) B A,B A,B We list two consequences of Theorems 5.2 and 5.3. 5.4. Theorem ([4], Theorem 10.6). Let A and B be n×n type-II matrices. If N contains a Schur invertible matrix G and H =Θ (G), then A,B A,B N =G◦N , N′ HT =N . A,B A A,B AT 5.5. Corollary ([4], Corollary 10.9). If (A,B) is an invertible Jones pair, then Θ (M)T =B−1Θ (M)B B A for all M ∈N . A NowwepresentanimportantapplicationofTheorems5.2and5.3,whichimplies that the Nomura algebras N , N and N′ have the same dimension. A A,B A,B 5.6. Theorem. Let (A,B) be an invertible Jones pair. Then N =A◦N , N′ BT =N A,B A A,B A and N′ =(N′ )T. A,B A,BT BOSE-MESNER ALGEBRAS ATTACHED TO INVERTIBLE JONES PAIRS 7 Proof. We get the first equality by letting G = A in Theorem 5.4. Since B = Θ (A), we have A,B N′ BT =N . A,B AT By Theorem 5.1, we have N =N and hence the second equality holds. AT A If we replace B by BT in the above equality, then we get N′ B =N . A,BT AT SincemultiplicationbyB isinjective,the dimensionsofN =N andN′ are A AT A,BT equal. Now we let G equal A and replace B by BT in Equation (5.1). We get N′ ⊆(N′ )TBT. BT A,BT ByTheorem5.1,N =N =N′ . SinceN andN′ havethesamedimension, A B BT A A,BT we have N =(N′ )TBT. A A,BT Thus N′ BT =(N′ )TBT, which leads to the last equality of the theorem. A,B A,BT 5.7. Corollary. Let (A,B) be an invertible Jones pair. Then N =N . A,B A,BT Moreover, if A is symmetric, then N =(N )T. A,B A,B Proof. Applying Theorem 5.6 to the invertible Jones pairs (A,B) and (A,BT) gives N =A◦N =N . A,B A A,BT Using the same equation, we have NT =AT ◦NT. Since N is closed under the A,B A A transpose and A is symmetric, we conclude that N =NT . A,B A,B 6. A Bose-Mesner Algebra of order 4n From now on, we assume that (A,B) is an invertible Jones pair and A is sym- metric. 6.1. Lemma. For each H in N , there exists a unique matrix K in (N )T A,B A,BT such that Θ (H)=Θ (KT)T. (6.1) A,B A,BT Proof. Existence follows directly from the last equality in Theorem 5.6, while uniqueness holds because Θ is an isomorphism. A,BT Given any matrix H in N , we say that the unique K in N′ satisfying A,B A,B Equation (6.1) is paired with H. 6.2. Lemma. For each H in N , K in N′ is paired with H if and only if KT A,B A,B is paired with HT. Moreover we have Θ (H ◦A)=Θ (KT ◦A). (6.2) A BT 8 ADACHANANDCHRISGODSIL Proof. MultiplyingeachsideofEquation(6.1)byn−1Θ (A)T =n−1Θ (A) A,B A,BT gives n−1Θ (H)Θ (A)T =n−1Θ (KT)TΘ (A). A,B A,B A,BT A,BT We apply Theorem 5.3 to both sides of the above equation to get Θ (H ◦AT)=Θ (KT ◦AT). A BT Since A is symmetric, we see that Equation (6.1) is equivalent to Equation (6.2). In addition, taking the transpose of both sides gives Θ (HT ◦A)=Θ (K◦A). A BT Therefore H and K satisfy Equation(6.1) if and only if HT and KT satisfy Equa- tion (6.1). For anyF ∈N andH,G∈N , we define the 4n×4nmatrix M(F,G,H) to A A,B be Θ (F)+H Θ (F)−H Θ (G) Θ (G) A A A,B A,B Θ (F)−H Θ (F)+H Θ (G) Θ (G)  A A A,B A,B , ΘA,B(GT)T ΘA,B(GT)T ΘB(−)(F)+K ΘB(−)(F)−K ΘA,B(GT)T ΘA,B(GT)T ΘB(−)(F)−K ΘB(−)(F)+K where Kis paired with H. We consider the space  B :={M(F,G,H):F ∈N and H,G∈N }. (6.3) A A,B Now we show that B is a Bose-Mesner algebra. It turns out that B contains the 4n×4ntype-IImatrixV definedatthebeginningofSection7anditisasubscheme ofN . ThisleadstothemainresultofthispaperwhichsaysthatV isaspinmodel V if and only if (A,B) is an invertible Jones pair. To convince ourselvesthat B is a Bose-Mesner algebra,we need to check that B contains the identity matrix I and the matrix of all ones J ; it is closed under 4n 4n the transpose;itis a commutative algebrawithrespectto matrix multiplication; it is closed under the Schur product. 6.3. Lemma. The vector space B contains I and J . 4n 4n Proof. The matrix K that is paired with 1I satisfies 2 n 1 1 Θ (KT)T =Θ ( I )= J . A,BT A,B 2 n 2 n SinceΘ isanisomorphism,weconcludethatK = 1I . NotethatΘ ( 1 J )= A,BT 2 n A 2n n 1I . Thus M( 1 J ,0,1I )=I belongs to B. 2 n 2n n 2 n 4n Since Θ (I )=Θ (I )=J , the matrix M(I ,I ,0)=J belongs to B. A n A,B n n n n 4n 6.4. Lemma. The vector space B is closed under transpose. Proof. Let M(F,G,H)∈B. Now M(F,G,H)T equals Θ (F)T +HT Θ (F)T −HT Θ (GT) Θ (GT) A A A,B A,B Θ (F)T −HT Θ (F)T +HT Θ (GT) Θ (GT)  A A A,B A,B . ΘA,B(G)T ΘA,B(G)T ΘB(−)(F)T +KT ΘB(−)(F)T −KT  ΘA,B(G)T ΘA,B(G)T ΘB(−)(F)T −KT ΘB(−)(F)T +KT   BOSE-MESNER ALGEBRAS ATTACHED TO INVERTIBLE JONES PAIRS 9 SinceN isclosedunderthetranspose,thematricesGT andHT belongtoN . A,B A,B It follows from Lemma 6.2 that KT is paired with HT. Moreover, Θ (F)T = A Θ (FT). As a result we conclude that A M(F,G,H)T =M(FT,GT,HT), and the vector space B is closed under the transpose. 6.5. Lemma. The vector space B is a commutative algebra under matrix multipli- cation. Proof. Let M =M(F,G,H) and M =M(F ,G ,H ) be any matrices in B. 1 1 1 1 By Theorem 5.3, we have Θ (G)Θ (GT)T =nΘ (G◦G ). A,B A,B 1 A 1 Hence the top left 2n×2n block of MM equals 1 2nΘ (F ◦F +G◦G )+2HH 2nΘ (F ◦F +G◦G )−2HH A 1 1 1 A 1 1 1 . 2nΘ (F ◦F +G◦G )−2HH 2nΘ (F ◦F +G◦G )+2HH A 1 1 1 A 1 1 1 (cid:18) (cid:19) Similarly, by Theorem 5.3 Θ (GT)TΘ (G ) = nΘ (GT ◦GT) A,B A,B 1 B 1 = nΘ (G◦G )T B 1 = nΘB(−)(G◦G1). Consequently the bottom right 2n×2n block of MM equals 1 2nΘB(−)(F ◦F1+G◦G1)+2KK1 2nΘB(−)(F ◦F1+G◦G1)−2KK1 , (cid:18)2nΘB(−)(F ◦F1+G◦G1)−2KK1 2nΘB(−)(F ◦F1+G◦G1)+2KK1(cid:19) where K and K are paired with H and H , respectively. Now we need to show 1 1 that KK is paired with HH . From Equation (6.1), we have 1 1 Θ (H)=Θ (KT)T and Θ (H )=Θ (KT)T. A,B A,BT A,B 1 A,BT 1 Therefore Θ (HH ) = Θ (H)◦Θ (H ) A,B 1 A,B A,B 1 = Θ (KT)T ◦Θ (KT)T A,BT A,BT 1 = Θ (KTKT)T. A,BT 1 Since N is commutative with respect to matrix multiplication, A,BT Θ (HH )=Θ ((KK )T)T. A,B 1 A,BT 1 We now consider the top right 2n×2n block of MM . Note that 1 2ΘA(F)ΘA,B(G1)+2ΘA,B(G)ΘB(−)(F1) = 2Θ (F)Θ (G )+2Θ (G)Θ (F )T. A A,B 1 A,B B 1 Applying Theorem 5.2 to each term, we get 2nΘ (F ◦G +G◦F ). A,B 1 1 Thus the top right 2n×2n block of MM is 1 2nΘ (F ◦G +G◦F ) 2nΘ (F ◦G +G◦F ) A,B 1 1 A,B 1 1 . 2nΘ (F ◦G +G◦F ) 2nΘ (F ◦G +G◦F ) A,B 1 1 A,B 1 1 (cid:18) (cid:19) 10 ADACHANANDCHRISGODSIL Consider the bottom left 2n×2n block of MM , we have 1 2ΘA,B(GT)TΘA(F1)+2ΘB(−)(F)ΘA,B(GT1)T = 2Θ (GT)TΘ (F )+2Θ (F)TΘ (GT)T. A,B A 1 B A,B 1 Since each of Θ and Θ commutes with the transpose, the above expression A B becomes 2Θ (GT)TΘ (FT)T +2Θ (FT)Θ (GT)T, A,B A 1 B A,B 1 which equals 2nΘ (FT ◦GT +GT ◦FT)T A,B 1 1 by Theorem 5.2. Hence the bottom left 2n×2n block of MM is 1 2nΘ (FT ◦GT +GT ◦FT)T 2nΘ (FT ◦GT +GT ◦FT)T A,B 1 1 A,B 1 1 . 2nΘ (FT ◦GT +GT ◦FT)T 2nΘ (FT ◦GT +GT ◦FT)T (cid:18) A,B 1 1 A,B 1 1 (cid:19) Now we conclude that MM =M(2nF ◦F +2nG◦G ,2nF ◦G +2nG◦F ,2HH ) 1 1 1 1 1 1 belongs to B. It follows from the commutativity of N = N that HH = H H and A,B A,BT 1 1 KK = K K. Therefore all four 2n×2n blocks of MM remain unchanged after 1 1 1 swapping F with F , G with G , H with H and K with K . Consequently the 1 1 1 1 matrices M and M commute. 1 6.6. Lemma. The algebra B is closed under the Schur product. Proof. Let M = M(F,G,H) and M = M(F ,G ,H ) be two matrices in B. 1 1 1 1 We want to write M ◦M as M(F′,G′,H′), for some F′ in N and G′ and H′ in 1 A N . If we divide M◦M into sixteenn×n blocks naturally, then the (1,1)- and A,B 1 (2,2)-blocks of M ◦M are equal to 1 Θ (F)◦Θ (F )+H ◦H +Θ (F)◦H +Θ (F )◦H A A 1 1 A 1 A 1 = (Θ (FF )+H ◦H )+(Θ (F)◦H +Θ (F )◦H). A 1 1 A 1 A 1 The (1,2)- and (2,1)-blocks of M ◦M are equal to 1 (Θ (FF )+H ◦H )−(Θ (F)◦H +Θ (F )◦H). A 1 1 A 1 A 1 The (3,3)- and (4,4)-blocks of M ◦M are equal to 1 (ΘB(−)(FF1)+K◦K1)+(ΘB(−)(F)◦K1+ΘB(−)(F1)◦K). The (3,4)- and (4,3)-blocks of M ◦M are equal to 1 (ΘB(−)(FF1)+K◦K1)−(ΘB(−)(F)◦K1+ΘB(−)(F1)◦K). To determine F′, we need to show that there exists Fˆ ∈N such that A H ◦H1 =ΘA(Fˆ) and K◦K1 =ΘB(−)(Fˆ), and F′ = FF +Fˆ. Now the matrix K is paired with H. Right-multiplying both 1 sides of Equation (6.1) by B(−)T yields Θ (H)B(−)T =Θ (KT)TB(−)T, A,B A,BT which is rewritten as Θ (H)Θ (A−1)T =Θ (KT)TΘ (A−1). A,B A,B A,BT A,BT