Nomura algebras of nonsymmetric Hadamard models
aa r X i v : . [ m a t h . C O ] O c t NOMURA ALGEBRAS OF NONSYMMETRICHADAMARD MODELS
TAKUYA IKUTA AND AKIHIRO MUNEMASA
Abstract.
We show that the Nomura algebra of the nonsymmet-ric Hadamard model coincides with the Bose–Mesner algebra of thedirected Hadamard graph. Introduction
Spin models for link invariants were introduced by Jones [7], andtheir connection to combinatorics was revealed first in [4]. Jaeger andNomura [6] constructed nonsymmetric spin models for link invariantsfrom Hadamard matrices, and showed that these models give link in-variants which depend nontrivially on link orientation. These mod-els are a modification of the Hadamard model originally constructedby Nomura [9]. Jaeger and Nomura also pointed out a similarity be-tween the association scheme of a Hadamard graph and the associationscheme containing their new nonsymmetric spin model.Nomura [10], and later Jaeger, Matsumoto and Nomura [5] intro-duced an algebra called the Nomura algebra of a type II matrix W ,and showed that this algebra coincides with the Bose–Mesner algebraof some self-dual association schemes when W is a spin model. By [5]the Nomura algebra of the Hadamard model coincides with the Bose–Mesner algebra of the corresponding Hadamard graph.The purpose of this paper is to determine the Nomura algebra of anonsymmetric Hadamard model to be the Bose–Mesner algebra of thecorresponding directed Hadamard graph. We also show that the di-rected Hadamard graph can be constructed from the ordinary Hadamardgraphs by means of a general method given by Klin, Muzychuk, Pech,Woldar and Zieschang [8].2. Preliminaries on Nomura algebras
Let X be a finite set with k elements. We denote by Mat X ( C )the algebra of square matrices with complex entries whose rows andcolumns are indexed by X . We also denote by Mat X ( C ∗ ) the subset Date : October 25, 2011. of Mat X ( C ) consisting of matrices all of whose entries are nonzero.For W ∈ Mat X ( C ) and x, y ∈ X , the ( x, y )-entry of W is denoted by W ( x, y ).A type II matrix is a matrix W ∈ Mat X ( C ∗ ) which satisfies the type II condition :(1) X x ∈ X W ( a, x ) W ( b, x ) = kδ a,b (for all a, b ∈ X ) . For a type II matrix W ∈ Mat X ( C ∗ ) and a, b ∈ X , we define a columnvector Y ab ∈ C X whose x -entry is given by Y ab ( x ) = W ( x, a ) W ( x, b ) . The Nomura algebra N ( W ) of W is defined by N ( W ) = { M ∈ Mat n ( C ∗ ) | Y ab is an eigenvector for M for all a, b ∈ X } . A type II matrix W is called a spin model if it satisfies the type IIIcondition : X x ∈ X W ( a, x ) W ( b, x ) W ( c, x ) = d W ( a, b ) W ( a, c ) W ( c, b ) (for all a, b, c ∈ X ) , where d = k . It is shown in [5, Theorem 11] that if W is a spin model,then N ( W ) is the Bose-Mesner algebra of some self-dual associationscheme.We shall associate with W an undirected graph G on the vertex set X × X . Given two column vectors T , T ′ indexed by X , we write h T, T ′ i for their Hermitian scalar product P x ∈ X T ( x ) T ′ ( x ). Two distinct ver-tices ( a, b ) , ( c, d ) will be adjacent in G iff h Y ab , Y cd i 6 = 0. For a subset C of X × X , we denote by A ( C ) the matrix in Mat X ( C ) with ( a, b )-entryequal to 1 if ( a, b ) ∈ C and to 0 otherwise. Then we have the following: Theorem 1. [5, Theorem 5(iii)]
Let C , . . . , C p be the connected compo-nents of G . Then the algebra N ( W T ) has a basis { A ( C i ) | i = 1 , . . . , p } . Hadamard graphs and directed Hadamard graphs
In this section, we define the adjacency matrices of Hadamard graphsand directed Hadamard graphs, and give the association schemes deter-mined by them. We refer the reader to [1, Theorem 1.8.1] for propertiesof Hadamard graphs, and to [2] for background materials in the theoryof association schemes.
OMURA ALGEBRAS OF NONSYMMETRIC HADAMARD MODELS 3
Let k be a positive integer, and let H ∈ Mat X ( C ) be a Hadamardmatrix of order k . We denote by I n the identity matrix of order n , andwe omit n if n = k . Let J ∈ Mat X ( C ) be the all 1’s matrix. We define A = I k ,A = ( J + H ) ( J − H )0 0 ( J − H ) ( J + H ) ( J + H T ) ( J − H T ) 0 0 ( J − H T ) ( J + H T ) 0 0 ,A = J − I J − I J − I J − I J − I J − I J − I J − I ,A = ( J − H ) ( J + H )0 0 ( J + H ) ( J − H ) ( J − H T ) ( J + H T ) 0 0 ( J + H T ) ( J − H T ) 0 0 ,A = I I I I . The matrices { A i } i =0 are the distance matrices of the Hadamard graphassociated to the Hadamard matrix H . Since the Hadamard graph isdistance-regular (see [1, Theorem 1.8.1]),(2) A = span { A , A , A , A , A } is closed under the matrix multiplication. This algebra is called theBose–Mesner algebra of the Hadamard graph.The directed Hadamard graph associated with a Hadamard matrix H is the digraph with adjacency matrix A ′ = ( J + H ) ( J − H )0 0 ( J − H ) ( J + H ) ( J − H T ) ( J + H T ) 0 0 ( J + H T ) ( J − H T ) 0 0 . The matrix A ′ generates a Bose–Mesner algebra A ′ with basis A , A ′ , A , A ′ = A ′ T , A (see [6]). We shall index the rows and columns of the matrices A and A ′ by X × ( Z / Z ) , in the order X × { (0 , } , X × { (0 , } , X × { (1 , } , X × { (1 , } . Then A is the Bose–Mesner algebra of the TAKUYA IKUTA AND AKIHIRO MUNEMASA association scheme on X × ( Z / Z ) with relations R = { (( a, α ) , ( a, α )) | ( a, α ) ∈ X × ( Z / Z ) } ,R = { (( a, α ) , ( b, β )) | ( α , β ) = (0 , , H ( a, b ) = ( − α + β }∪ { (( a, α ) , ( b, β )) | ( α , β ) = (1 , , H ( b, a ) = ( − α + β } ,R = { (( a, α ) , ( b, β )) | α = β , a = b } ,R = { (( a, α ) , ( b, β )) | ( α , β ) = (0 , , H ( a, b ) = ( − α + β +1 }∪ { (( a, α ) , ( b, β )) | ( α , β ) = (1 , , H ( b, a ) = ( − α + β +1 } ,R = { (( a, α ) , ( a, β )) | α = β , α = β } . Also, A ′ is the Bose–Mesner algebra of an association scheme on X × ( Z / Z ) with relations R , R ′ , R , R ′ , R , where R ′ = { (( a, α ) , ( b, β )) | ( α , β ) = (0 , , H ( a, b ) = ( − α + β }∪ { (( a, α ) , ( b, β )) | ( α , β ) = (1 , , H ( b, a ) = ( − α + β +1 } ,R ′ = { (( a, α ) , ( b, β )) | ( α , β ) = (0 , , H ( a, b ) = ( − α + β +1 }∪ { (( a, α ) , ( b, β )) | ( α , β ) = (1 , , H ( b, a ) = ( − α + β } . Let Z = X × { } × Z / Z ,Z = X × { } × Z / Z ,Z = Z ∪ Z ,R i = R i ∩ ( Z × Z ) ,R i = R i ∩ ( Z × Z ) . Let R = { R i | i = 0 , . . . , } ∪ { R i | i = 0 , . . . , } . Then R is a coherent configuration in the sense of [3]. Let R ′ = R ∪ R , R ′ = R ∪ R . Then R λi R µj = δ i + λ mod 2 ,µ X k ≡ i + j (mod 2) p kij R λk . It follows that the permutation ρ of R defined by ρ ( R δi ) = R − δ if i = 1, R − δ if i = 3, R − δi otherwiseis an algebraic automorphism of the coherent configuration R in thesense of [8]. Since the relations R ′ = { R , R ′ , R , R ′ , R } are obtained OMURA ALGEBRAS OF NONSYMMETRIC HADAMARD MODELS 5 by fusing ρ -orbits, the fact that R ′ forms an association scheme followsalso from the general theory given in [8, Subsection 2.6].4. Symmetric and nonsymmetric Hadamard models
Throughout this section, we assume that k is an integer with k ≥ u be a complex number satisfying(3) k = ( u + u − ) A Potts model A ∈ Mat X ( C ∗ ) is defined as(4) A = u I − u − ( J − I ) . Let H ∈ Mat X ( C ) be a Hadamard matrix of order k . In [9], [6], thefollowing two spin models are given: W = A A ωH − ωHA A − ωH ωHωH T − ωH T A A − ωH T ωH T A A (5) = u A + ωA − u − A − ωA + u A ,W ′ = A A ξH − ξHA A − ξH ξH − ξH T ξH T A AξH T − ξH T A A (6) = u A + ξA ′ − u − A − ξA ′ + u A , where ω is a 4-th root of unity, ξ is a primitive 8-th root of unity,respectively. We index the rows and columns of the matrices (5), (6)by X × ( Z / Z ) as in Section 3. The spin models (5) and (6) are called aHadamard model and a nonsymmetric Hadamard model, respectively.From [5, Subsection 5.5] we have(7) N ( W ) = A . TAKUYA IKUTA AND AKIHIRO MUNEMASA
In order to determine the Nomura algebra N ( W ′ ), we consider thenormalized version of the matrices (5), (6):˜ W = A A H − HA A − H HH T − H T A A − H T H T A A , (8) ˜ W ′ = A A H − HA A − H HiH T − iH T A A − iH T iH T A A , (9)where i = − ξ is a primitive 4th root of unity. Then N ( W ) = N ( ˜ W )and N ( W ′ ) = N ( ˜ W ′ ), since˜ W = I I ωI ωI W I I ω − I ω − I if ω = 1, ˜ W = I I ωI ωI W I I ω − I ω − I if ω = −
1, and˜ W ′ = I I ξI ξI W ′ I I ξ − I ξ − I (see [5, Propositions 2 and 3]).Define column vectors Y αβab , Y ′ αβab whose x -entries are given by Y αβab ( x ) = ˜ W ( x, ( a, α ))˜ W ( x, ( b, β )) , (10) Y ′ αβab ( x ) = ˜ W ′ ( x, ( a, α ))˜ W ′ ( x, ( b, β ))(11)for ( a, α ) , ( b, β ) ∈ X × ( Z / Z ) , x ∈ X × ( Z / Z ) . OMURA ALGEBRAS OF NONSYMMETRIC HADAMARD MODELS 7
Lemma 2.
Let G and G ′ be the graphs with the same vertex set ( X × ( Z / Z ) ) , where two distinct vertices (( a, α ) , ( b, β )) , (( a ′ , α ′ ) , ( b ′ , β ′ )) are adjacent whenever h Y αβab , Y α ′ β ′ a ′ b ′ i 6 = 0 , h Y ′ αβab , Y ′ α ′ β ′ a ′ b ′ i 6 = 0 , respectively. Let K j ( j = 1 , . . . , p ) (resp. K ′ j ( j = 1 , . . . , p ′ ) ) be theconnected components of G (resp. G ′ ). Then N ( W ) (resp. N ( W ′ ) ) isspanned by { A ( K j ) | j = 1 , . . . , p } (resp. { A ( K ′ j ) | j = 1 , . . . , p ′ } ).Proof. By [5, Section 5.5], (7) holds regardless of the value of ω , so wehave N ( W ) = N ( ˜ W ). Since ˜ W = ˜ W T , the result for N ( W ) followsimmediately from Theorem 1.Since ˜ W ′ T = (cid:20) I k − ξ − I k (cid:21) W ′ (cid:20) I k − ξI k (cid:21) , we have N ( W ′ ) = N ( ˜ W ′ T ) by [5, Proposition 2]. Thus, the result for N ( W ′ ) follows also from Theorem 1. (cid:3) Let D = I I iI iI ∈ Mat X × ( Z / Z ) ( C ) . Lemma 3.
For ( a, α ) , ( b, β ) ∈ X × ( Z / Z ) , Y ′ αβab = Y αβab if α = β , DY αβab if ( α , β ) = (0 , , D − Y αβab if ( α , β ) = (1 , .Proof. Immediate from the definitions (8), (9), (10) and (11). (cid:3)
Lemma 4.
Let τ denote the permutation of ( Z / Z ) defined by τ ( α , α ) = ( α , α + α ) , and let σ denote the permutation of ( X × ( Z / Z ) ) defined by σ (( a, α ) , ( b, β )) = (( a, τ ( α )) , ( b, β )) . Then σ ( R ) = R ′ and σ ( R ) = R ′ .Proof. Immediate from the definitions of R , R , R ′ and R ′ . (cid:3) Lemma 5.
For ( a, α ) , ( b, β ) ∈ X × ( Z / Z ) , Y τ ( α ) βab = ( − D ) α Y αβab . TAKUYA IKUTA AND AKIHIRO MUNEMASA
Proof.
Since τ ( α ) = α when α = 0, the result holds in this case. If α = 1, then τ ( α ) = (1 , α ). Since the ( a, α )-column and ( a, τ ( α ))-column of ˜ W differ by the left multiplication by − D , the results holdsin this case as well. (cid:3) Lemma 6. If (( a, α ) , ( b, β )) , (( a ′ , α ′ ) , ( b ′ , β ′ )) ∈ R ∪ R , then h Y ′ τ ( α ) βab , Y ′ τ ( α ′ ) β ′ a ′ b ′ i = ( − α + α ′ h Y αβab , Y α ′ β ′ a ′ b ′ i . Proof.
Using Lemmas 3 and 5, we have h Y ′ τ ( α ) βab , Y ′ τ ( α ′ ) β ′ a ′ b ′ i = h DY τ ( α ) βab , DY τ ( α ′ ) β ′ a ′ b ′ i if ( τ ( α ) , β ) = ( τ ( α ′ ) , β ′ ) = (0 , h D − Y τ ( α ) βab , D − Y τ ( α ′ ) β ′ a ′ b ′ i if ( τ ( α ) , β ) = ( τ ( α ′ ) , β ′ ) = (1 , h DY τ ( α ) βab , D − Y τ ( α ′ ) β ′ a ′ b ′ i if ( τ ( α ) , β ) = ( β ′ , τ ( α ′ ) ) = (0 , h D − Y τ ( α ) βab , DY τ ( α ′ ) β ′ a ′ b ′ i if ( τ ( α ) , β ) = ( β ′ , τ ( α ′ ) ) = (1 , ( h Y τ ( α ) βab , Y τ ( α ′ ) β ′ a ′ b ′ i if ( α , β ) = ( α ′ , β ′ ), h D Y τ ( α ) βab , Y τ ( α ′ ) β ′ a ′ b ′ i otherwise= ( h ( − D ) α Y αβab , ( − D ) α ′ Y α ′ β ′ a ′ b ′ i if α = α ′ , h D ( − D ) α Y αβab , ( − D ) α ′ Y α ′ β ′ a ′ b ′ i otherwise= ( h Y αβab , Y α ′ β ′ a ′ b ′ i if α = α ′ , −h Y αβab , Y α ′ β ′ a ′ b ′ i otherwise= ( − α + α ′ h Y αβab , Y α ′ β ′ a ′ b ′ i . (cid:3) Theorem 7.
The Nomura algebra N ( W ′ ) of the spin model W ′ coin-cides with the Bose–Mesner algebra A ′ of the directed Hadamard graph.Proof. Since u = 1 or | u | >
1, the coefficients { ξ, − u − , − ξ, u } of W ′ in A ′ , A , A ′ , A are pairwise distinct. Since W ′ ∈ N ( W ′ ) by [5,Proposition 9], we obtain A ′ , A , A ′ , A ∈ N ( W ′ ). By Lemma 2, thisimplies that each of R ′ , R , R ′ , R is a union of connected componentsof G ′ .Since N ( W ) = A , Lemma 2 implies that R , R , . . . , R are theconnected components of G . Observe R ∪ R ∪ R = { (( a, α ) , ( b, β )) ∈ ( X × ( Z / Z ) ) | α = β } . By Lemma 3, we have Y αβab = Y ′ αβab for (( a, α ) , ( b, β )) ∈ R ∪ R ∪ R . OMURA ALGEBRAS OF NONSYMMETRIC HADAMARD MODELS 9
This implies that two graphs G and G ′ have the same set of edges on R ∪ R ∪ R . Thus R , R and R are connected.By Lemmas 4 and 6, there is an isomorphism σ from the subgraphof G induced by R ∪ R , to the subgraph of G ′ induced by R ′ ∪ R ′ ,satisfying σ ( R ) = R ′ and σ ( R ) = R ′ . Since R and R are connectedcomponents of G , R ′ and R ′ are connected.Therefore, we have shown that R , R ′ , R , R ′ , R are the connectedcomponents of G ′ . The result now follows from Lemma 2. (cid:3) Remark 8.
The condition of Theorem 7 does not hold when k = 1 , N ( W ) = N ( W ′ ) is the Bose–Mesneralgebra of the group association scheme of Z / Z when k = 1, and that N ( W ) ∼ = N ( W ′ ) is the Bose–Mesner algebra of the group associationscheme of Z / Z when k = 2. Acknowledgements.
The authors would like to thank Mitsugu Hi-rasaka for bringing the article [8] to the authors’ attention, and ananonymous referee for careful reading of the manuscript.
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