A diagram associated with the subconstituent algebra of a distance-regular graph
aa r X i v : . [ m a t h . C O ] D ec A diagram associated with the subconstituentalgebra of a distance-regular graph
Supalak Sumalroj
Department of Mathematics, Silpakorn University, Nakhon Pathom, Thailandsumalroj [email protected]
Abstract
In this paper we consider a distance-regular graph Γ. Fix a vertex x ofΓ and consider the corresponding subconstituent algebra T . The algebra T is the C -algebra generated by the Bose-Mesner algebra M of Γ and the dualBose-Mesner algebra M ∗ of Γ with respect to x . We consider the subspaces M, M ∗ , M M ∗ , M ∗ M, M M ∗ M, M ∗ M M ∗ , . . . along with their intersections andsums. In our notation, M M ∗ means Span { RS | R ∈ M, S ∈ M ∗ } , and so on.We introduce a diagram that describes how these subspaces are related. Wedescribe in detail that part of the diagram up to M M ∗ + M ∗ M . For eachsubspace U shown in this part of the diagram, we display an orthogonal basisfor U along with the dimension of U . For an edge U ⊆ W from this part of thediagram, we display an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement.Keywords : subconstituent algebra, Terwilliger algebra, distance-regular graphMath. Subj. Class.: 05E30 In this paper we consider a distance-regular graph Γ. Fix a vertex x of Γ and considerthe corresponding subconstituent algebra (or Terwilliger algebra) T [35]. The algebra T is the C -algebra generated by the Bose-Mesner algebra M of Γ and the dual Bose-Mesner algebra M ∗ of Γ with respect to x . The algebra T is finite-dimensional andsemisimple [35]. So it is natural to compute the irreducible T -modules. These modulesare important in the study of hypercubes [14, 26], dual polar graphs [20, 38], spinmodels [6, 10], codes [13, 28], the bipartite property [3, 4, 9, 16, 21, 22, 23, 25, 27],the almost-bipartite property [5, 8, 17], the Q -polynomial property [3, 7, 11, 12, 18,19, 27, 33], and the thin property [15, 24, 30, 31, 32, 34, 36, 37].1In this paper we discuss the algebra T using a difference approach. We considerthe subspaces M, M ∗ , M M ∗ , M ∗ M, M M ∗ M, M ∗ M M ∗ , . . . along with their intersec-tions and sums; see Figure 1. We describe the diagram of Figure 1 up to M M ∗ + M ∗ M .For each subspace U shown in this part of the diagram, we display an orthogonal basisfor U along with the dimension of U . For an edge U ⊆ W from this part of the dia-gram, we display an orthogonal basis for the orthogonal complement of U in W alongwith the dimension of this orthogonal complement. Our main results are summarizedin Theorems 36, 37. In the last part of the paper we summarize what is known aboutthe part of diagram above M M ∗ + M ∗ M , and we give some open problems. Let X denote a nonempty finite set. Let Mat X ( C ) denote the C -algebra consisting ofthe matrices whose rows and columns are indexed by X and whose entries are in C .For B ∈ Mat X ( C ) let B , B t , and tr ( B ) denote the complex conjugate, the transpose,and the trace of B , respectively. We endow Mat X ( C ) with the Hermitean innerproduct h , i such that h R, S i = tr ( R t S ) for all R, S ∈ Mat X ( C ). The inner product h , i is positive definite. Let U, V denote subspaces of Mat X ( C ) such that U ⊆ V . The orthogonal complement of U in V is defined by U ⊥ = { v ∈ V |h v, u i = 0 for all u ∈ U } .Let Γ = ( X, E ) denote a finite, undirected, connected graph, without loops ormultiple edges, with vertex set X and edge set E . Let ∂ denote the shortest path-length distance function for Γ. Define the diameter D := max { ∂ ( x, y ) | x, y ∈ X } .For a vertex x ∈ X and an integer i ≥ i ( x ) = { y ∈ X | ∂ ( x, y ) = i } . Fornotational convenience abbreviate Γ( x ) = Γ ( x ). For an integer k ≥
0, we say thatΓ is regular with valency k whenever | Γ( x ) | = k for all x ∈ X . We say that Γ is distance-regular whenever for all integers h, i, j (0 ≤ h, i, j ≤ D ) and x, y ∈ X with ∂ ( x, y ) = h , the number p hij := | Γ i ( x ) ∩ Γ j ( y ) | is independent of x and y . The integers p hij are called the intersection numbers of Γ.From now on assume that Γ is distance-regular with diameter D ≥
3. We abbreviate k i := p ii (0 ≤ i ≤ D ). For 0 ≤ i ≤ D let A i denote the matrix in Mat X ( C ) with( x, y )-entry ( A i ) xy = ( ∂ ( x, y ) = i, ∂ ( x, y ) = i, x, y ∈ X. We call A i the i -th distance matrix of Γ. We call A = A the adjacency matrix of Γ.Observe that A i is real and symmetric for 0 ≤ i ≤ D . Note that A = I is the identitymatrix in Mat X ( C ). Observe that P Di =0 A i = J , where J is the all-ones matrix inMat X ( C ). Observe that for 0 ≤ i, j ≤ D , A i A j = D X h =0 p hij A h . (1)For integers h, i, j (0 ≤ h, i, j ≤ D ) we have p h j = δ hj (2) p ij = δ ij k i (3)Let M denote the subalgebra of Mat X ( C ) generated by A . By [2, p. 44] thematrices A , A , ..., A D form a basis for M . We call M the Bose-Mesner algebra of
Γ. By [1, p. 59, 64], M has a basis E , E , ..., E D such that (i) E = | X | − J ; (ii) P Di =0 E i = I ; (iii) E ti = E i (0 ≤ i ≤ D ); (iv) E i = E i (0 ≤ i ≤ D ); (v) E i E j = δ ij E i (0 ≤ i, j ≤ D ). The matrices E , E , ..., E D are called the primitive idempotents of Γ,and E is called the trivial idempotent. For 0 ≤ i ≤ D let m i denote the rank of E i .For 0 ≤ i ≤ D let θ i denote an eigenvalue of A associated with E i . Let λ denote anindeterminate. Define polynomials { u i } Di =0 in C [ λ ] by u = 1, u = λ/k , and λu i = c i u i − + a i u i + b i u i +1 (1 ≤ i ≤ D − . By [2, p. 131, 132], A j = k j D X i =0 u j ( θ i ) E i (0 ≤ j ≤ D ) , (4) E j = | X | − m j D X i =0 u i ( θ j ) A i (0 ≤ j ≤ D ) . (5)Since E i E j = δ ij E i and by (4), (5) we have A j E i = k j u j ( θ i ) E i = E i A j (0 ≤ i, j ≤ D ).By [1, Theorem 3.5] we have the orthogonality relations D X i =0 u i ( θ r ) u i ( θ s ) k i = δ rs m − r | X | (0 ≤ r, s ≤ D ) , (6) D X r =0 u i ( θ r ) u j ( θ r ) m r = δ ij k − i | X | (0 ≤ i, j ≤ D ) . (7)We recall the Krein parameters of Γ. Let ◦ denote the entry-wise multiplicationin Mat X ( C ). Note that A i ◦ A j = δ ij A i for 0 ≤ i, j ≤ D . So M is closed under ◦ . By[2, p. 48], there exist scalars q hij ∈ C such that E i ◦ E j = | X | − D X h =0 q hij E h (0 ≤ i, j ≤ D ) . (8)We call the q hij the Krein parameters of Γ. By [2, Proposition 4.1.5], these parametersare real and nonnegative for 0 ≤ h, i, j ≤ D .We recall the dual Bose-Mesner algebra of Γ. Fix a vertex x ∈ X . For0 ≤ i ≤ D let E ∗ i = E ∗ i ( x ) denote the diagonal matrix in Mat X ( C ) with ( y, y )-entry( E ∗ i ) yy = ( ∂ ( x, y ) = i, ∂ ( x, y ) = i, y ∈ X. We call E ∗ i the i -th dual idempotent of Γ with respect to x . Observe that (i) P Di =0 E ∗ i = I ; (ii) E ∗ ti = E ∗ i (0 ≤ i ≤ D ); (iii) E ∗ i = E ∗ i (0 ≤ i ≤ D ); (iv) E ∗ i E ∗ j = δ ij E ∗ i (0 ≤ i, j ≤ D ). By construction E ∗ , E ∗ , ..., E ∗ D are linearly independent. Let M ∗ = M ∗ ( x )denote the subalgebra of Mat X ( C ) with basis E ∗ , E ∗ , ..., E ∗ D . We call M ∗ the dualBose-Mesner algebra of Γ with respect to x .We now recall the dual distance matrices of Γ. For 0 ≤ i ≤ D let A ∗ i = A ∗ i ( x )denote the diagonal matrix in Mat X ( C ) with ( y, y )-entry( A ∗ i ) yy = | X | ( E i ) xy y ∈ X. (9)We call A ∗ i the dual distance matrix of Γ with respect to x and E i . By [35, p.379], the matrices A ∗ , A ∗ , ..., A ∗ D form a basis for M ∗ . Observe that (i) A ∗ = I ; (ii) P Di =0 A ∗ i = | X | E ∗ ; (iii) A ∗ ti = A ∗ i (0 ≤ i ≤ D ); (iv) A ∗ i = A ∗ i (0 ≤ i ≤ D ); (v) A ∗ i A ∗ j = P Dh =0 q hij A ∗ h (0 ≤ i, j ≤ D ). From (4), (5) we have A ∗ j = m j D X i =0 u i ( θ j ) E ∗ i (0 ≤ j ≤ D ) , (10) E ∗ j = | X | − k j D X i =0 u j ( θ i ) A ∗ i (0 ≤ j ≤ D ) . (11) T Let T denote the subalgebra of Mat X ( C ) generated by M, M ∗ . The algebra T iscalled the subconstituent algebra (or Terwilliger algebra ) [35]. In order to describe T , we consider how M, M ∗ are related. We will use the following notation. Forany two subspaces R , S of Mat X ( C ) we define RS = Span { RS | R ∈ R , S ∈ S} .Consider the subspaces M, M ∗ , M M ∗ , M ∗ M, M M ∗ M, M ∗ M M ∗ , . . . along with theirintersections and sums. To describe the inclusions among the resulting subspaces wedraw a diagram; see Figure 1. In this diagram, a line segment that goes upward from U to W means that W contains U . M ∗ M M ∗ M M ∗ M M M ∗ M ∩ M ∗ M M ∗ M M ∗ M + M ∗ M M ∗ T C IM M ∗ + M ∗ M M ∗ MM M ∗ M M ∗ ∩ M ∗ MM + M ∗ M ∗ M M ∩ M ∗ Figure 1: Inclusion diagramConsider the above diagram. For each subspace U shown in the diagram, wedesire to find an orthogonal basis for U and the dimension of U . Also, for each edge U ⊆ W shown in the diagram, we desire to find an orthogonal basis for the orthogonalcompliment of U in W along with the dimension of this orthogonal compliment. Weaccomplish these goals for that part of the diagram up to M M ∗ + M ∗ M . Our mainresults are summarized in Theorems 36, 37. Before we get started, we recall a fewinner product formulas. Lemma 1. (See [ , Lemma 3 . . ) For ≤ h, i, j, r, s, t ≤ D ,(i) h E ∗ i A j E ∗ h , E ∗ r A s E ∗ t i = δ ir δ js δ ht k h p hij ,(ii) h E i A ∗ j E h , E r A ∗ s E t i = δ ir δ js δ ht m h q hij . The following result is well-known.
Corollary 2.
For ≤ h, i, j ≤ D ,(i) E ∗ i A h E ∗ j = 0 if and only if p hij = 0 ,(ii) E i A ∗ h E j = 0 if and only if q hij = 0 . Lemma 3. (See [ , Lemma 10] . ) For ≤ h, i, j, r, s, t ≤ D , h A i E ∗ j A h , A r E ∗ s A t i = D X ℓ =0 k ℓ p ℓir p ℓjs p ℓht . M + M ∗ Our goal in this section is to analyze the inclusion diagram up to M + M ∗ . Lemma 4.
For ≤ i ≤ D ,(i) tr ( A i ) = δ i | X | ,(ii) tr ( E i ) = m i ,(iii) tr ( A ∗ i ) = δ i | X | ,(iv) tr ( E ∗ i ) = k i .Proof. ( i ) Follows from the definition of A i .( ii ) Since E i is diagonalizable, we have tr ( E i ) = rank ( E i ) = m i .( iii ) Follows from the definition of A ∗ i .( iv ) Follows from the definition of E ∗ i . Lemma 5.
For ≤ i, j ≤ D ,(i) h A i , A j i = δ ij k i | X | ,(ii) h E i , E j i = δ ij m i ,(iii) h A ∗ i , A ∗ j i = δ ij m i | X | ,(iv) h E ∗ i , E ∗ j i = δ ij k i . Proof. ( i ) Use (1) and Lemma 4.( ii ) By Lemma 4 and since E i E j = δ ij E i .( iii ) , ( iv ) Similar to the proofs of ( i ) , ( ii ). Lemma 6.
Each of the following is an orthogonal basis for M : { A i } Di =0 , { E i } Di =0 . Moreover, each of the following is an orthogonal basis for M ∗ : { A ∗ i } Di =0 , { E ∗ i } Di =0 . Proof.
Immediate from Lemma 5.
Lemma 7.
For ≤ i, j ≤ D , h A i , A ∗ j i = δ i δ j | X | k i .Proof. Observe that h A i , A ∗ j i = h A i A ∗ A , A A ∗ j A i . By Lemma 3 and (6), (10), theresult follows.Recall that A = I = A ∗ . Lemma 8.
The following is an orthogonal basis for M + M ∗ : A D , . . . , A , I, A ∗ . . . , A ∗ D . Proof.
Immediate from Lemmas 5 and 7.
Lemma 9. dim ( M + M ∗ ) = 2 D + 1 .Proof. Immediate from Lemma 8.
Lemma 10.
We have M ∩ M ∗ = C I and dim ( M ∩ M ∗ ) = 1 .Proof. Observe that I ∈ M ∩ M ∗ . By linear algebra, we have dim ( M ∩ M ∗ ) = dim ( M ) + dim ( M ∗ ) − dim ( M + M ∗ ). By construction dim ( M ) = D + 1, dim ( M ∗ ) = D + 1. By this and Lemma 9, dim ( M ∩ M ∗ ) = 1. The result follows. Lemma 11.
The following ( i ) – ( iv ) hold:(i) The matrices { A i } Di =1 form an orthogonal basis for the orthogonal complementof M ∩ M ∗ in M .(ii) The matrices { A ∗ i } Di =1 form an orthogonal basis for the orthogonal complementof M ∩ M ∗ in M ∗ .(iii) The matrices { A i } Di =1 form an orthogonal basis for the orthogonal complementof M ∗ in M + M ∗ . (iv) The matrices { A ∗ i } Di =1 form an orthogonal basis for the orthogonal complementof M in M + M ∗ .Proof. Follows from definitions of
M, M ∗ along with Lemmas 8 and 10. Lemma 12.
Each of the following subspaces has dimension D : ( M ∩ M ∗ ) ⊥ ∩ M, ( M ∩ M ∗ ) ⊥ ∩ M ∗ , ( M ∗ ) ⊥ ∩ ( M + M ∗ ) , M ⊥ ∩ ( M + M ∗ ) . Proof.
Immediate from Lemma 11.
M M ∗ + M ∗ M Our goal in this section is to analyze the inclusion diagram from M + M ∗ up to M M ∗ + M ∗ M . Lemma 13.
For ≤ i, j, r, s ≤ D ,(i) h A i A ∗ j , A ∗ r A s i = δ is δ jr | X | k i m j u i ( θ j ) ,(ii) h A i A ∗ j , A r A ∗ s i = δ ir δ js | X | k i m j .Proof. ( i ) By (5), (9) and Lemma 5, we obtain h A i A ∗ j , A ∗ r A s i = tr (( A i A ∗ j ) t ( A ∗ r A s ))= tr ( A ∗ j A i A ∗ r A s )= X y ∈ X X z ∈ X ( A ∗ j ) yy ( A i ) yz ( A ∗ r ) zz ( A s ) zy = | X | X y ∈ X X z ∈ X ( E j ) xy ( A i ) yz ( E r ) xz ( A s ) zy = | X | X y ∈ X X z ∈ X X x ∈ X ( E j ) xy ( A i ) yz ( E r ) xz ( A s ) zy = | X | tr ( E j E r ( A i ◦ A s ))= | X | tr (( E j E r ) t ( A i ◦ A s ))= | X |h E j E r , A i ◦ A s i = δ is δ jr | X |h E j , A i i = δ is δ jr m j D X h =0 u h ( θ j ) h A h , A i i = δ is δ jr m j D X h =0 u h ( θ j ) δ hi k i | X | = δ is δ jr | X | k i m j u i ( θ j ) . ( ii ) By Lemma 3 and (2), (3), (6), (10) we obtain h A i A ∗ j , A r A ∗ s i = h A i A ∗ j A , A r A ∗ s A i = m j m s D X h =0 u h ( θ j ) D X ℓ =0 u ℓ ( θ s ) h A i E ∗ h A , A r E ∗ ℓ A i = m j m s D X h =0 u h ( θ j ) D X ℓ =0 u ℓ ( θ s ) D X t =0 k t p tir p thℓ p t = m j m s D X h =0 u h ( θ j ) D X ℓ =0 u ℓ ( θ s ) k p ir p hℓ = δ ir k i m j m s D X h =0 u h ( θ j ) u h ( θ s ) k h = δ ir k i m j m s δ js m − j | X | = δ ir δ js | X | k i m j . Lemma 14.
The following ( i ) , ( ii ) hold:(i) The matrices { A i A ∗ j | ≤ i, j ≤ D } form an orthogonal basis for M M ∗ .(ii) The matrices { A ∗ j A i | ≤ i, j ≤ D } form an orthogonal basis for M ∗ M .Proof. Immediate from Lemma 13.
Lemma 15.
Each of the following subspaces has dimension ( D + 1) : M M ∗ , M ∗ M. Proof.
Immediate from Lemma 14.
Lemma 16.
We have
M M ∗ + M ∗ M = D X i =0 D X j =0 Span ( A i A ∗ j , A ∗ j A i ) (orthogonal direct sum).Proof. Immediate from Lemma 13.
Corollary 17.
We have dim ( M M ∗ + M ∗ M ) = D X i =0 D X j =0 dim ( Span ( A i A ∗ j , A ∗ j A i )) . Proof.
Immediate from Lemma 16.0
Definition 18.
For 0 ≤ i, j ≤ D let H i,j denote the 2 × A i A ∗ j , A ∗ j A i . Lemma 19.
For ≤ i, j ≤ D , H i,j = | X | k i m j u i ( θ j ) u i ( θ j ) 1 . Proof.
Immediate from Lemma 13 and Definition 18.
Lemma 20.
For ≤ i, j ≤ D we have det ( H i,j ) = | X | k i m j (1 − ( u i ( θ j )) ) . Proof.
Immediate from Lemma 19.
Corollary 21.
For ≤ i, j ≤ D , det ( H i,j ) = 0 if and only if u i ( θ j ) = ± .Proof. Immediate from Lemma 20.
Lemma 22.
The following elements are orthogonal: A i A ∗ j + A ∗ j A i , A i A ∗ j − A ∗ j A i . Moreover || A i A ∗ j + A ∗ j A i || = 2 | X | k i m j (1 + u i ( θ j )) , || A i A ∗ j − A ∗ j A i || = 2 | X | k i m j (1 − u i ( θ j )) . Proof.
Immediate from Lemma 19.
Lemma 23.
The following ( i ) – ( iii ) hold for ≤ i, j ≤ D :(i) Assume u i ( θ j ) = 1 . Then A i A ∗ j = A ∗ j A i and this common value is nonzero.(ii) Assume u i ( θ j ) = − . Then A i A ∗ j = − A ∗ j A i and this common value is nonzero.(iii) Assume u i ( θ j ) = ± . Then A i A ∗ j , A ∗ j A i are linearly independent.Proof. ( i ) , ( ii ) Immediate from Lemma 22.( iii ) Immediate from Lemma 20. Lemma 24.
For ≤ i, j ≤ D we give an orthogonal basis for Span ( A i A ∗ j , A ∗ j A i ) .case orthogonal basis dimension u i ( θ j ) = ± A i A ∗ j u i ( θ j ) = ± A i A ∗ j + A ∗ j A i , A i A ∗ j − A ∗ j A i Proof.
Follows from Definition 18 and Lemmas 19, 23.
Corollary 25.
The following is an orthogonal basis for
M M ∗ + M ∗ M : { A i A ∗ j + A ∗ j A i , A i A ∗ j − A ∗ j A i | ≤ i, j ≤ D, u i ( θ j ) = ± }∪{ A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } . Proof.
Immediate from Lemmas 16 and 24.
Definition 26.
Define an integer P as follows: P = |{ ( i, j ) | ≤ i, j ≤ D, u i ( θ j ) = ± }| . Lemma 27. P = 0 if and only if Γ is primitive.Proof. Immediate from Definition 26 and [2, Proposition 4.4.7].
Lemma 28. dim ( M M ∗ + M ∗ M ) = 2 D + 2 D + 1 − P .Proof. Immediate from Corollary 25 and Definition 26.
Lemma 29. dim ( M M ∗ ∩ M ∗ M ) = 2 D + 1 + P .Proof. By linear algebra, we have dim ( M M ∗ ∩ M ∗ M ) = dim ( M M ∗ ) + dim ( M ∗ M ) − dim ( M M ∗ + M ∗ M ) . By Lemmas 15 and 28, the result follows.
Lemma 30.
The following is an orthogonal basis for
M M ∗ ∩ M ∗ M : { A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } . Proof.
Immediate from Corollary 21 and Lemma 29.
Lemma 31.
The matrices { A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } form an orthogonalbasis for the orthogonal complement of M + M ∗ in M M ∗ ∩ M ∗ M .Proof. Follows from Lemmas 8 and 30.
Lemma 32.
The following ( i ) , ( ii ) hold:(i) The matrices { A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } form an orthogonal basis for theorthogonal complement of M M ∗ ∩ M ∗ M in M M ∗ .(ii) The matrices { A ∗ j A i | ≤ i, j ≤ D, u i ( θ j ) = ± } form an orthogonal basis for theorthogonal complement of M M ∗ ∩ M ∗ M in M ∗ M .Proof. Follows from Lemmas 14 and 30.2
Lemma 33.
The following ( i ) , ( ii ) hold:(i) The matrices { u i ( θ j ) A i A ∗ j − A ∗ j A i | ≤ i, j ≤ D, u i ( θ j ) = ± } form an orthogonalbasis for the orthogonal complement of M M ∗ in M M ∗ + M ∗ M .(ii) The matrices { A i A ∗ j − u i ( θ j ) A ∗ j A i | ≤ i, j ≤ D, u i ( θ j ) = ± } form an orthogonalbasis for the orthogonal complement of M ∗ M in M M ∗ + M ∗ M .Proof. Follows from Lemma 14 and Corollary 25.
Lemma 34.
The following subspace has dimension P : ( M + M ∗ ) ⊥ ∩ ( M M ∗ ∩ M ∗ M ) . Proof.
Immediate from Definition 26 and Lemma 31.
Lemma 35.
Each of the following subspaces has dimension D − P : ( M M ∗ ∩ M ∗ M ) ⊥ ∩ M M ∗ , ( M M ∗ ∩ M ∗ M ) ⊥ ∩ M ∗ M, ( M M ∗ ) ⊥ ∩ ( M M ∗ + M ∗ M ) , ( M ∗ M ) ⊥ ∩ ( M M ∗ + M ∗ M ) . Proof.
Immediate from Definition 26 and Lemmas 32, 33.
We summarize our results in the following theorems.
Theorem 36.
In each row of the table below we describe a subspace U in the diagramof Figure 1. We give an orthogonal basis for U along with the dimension of U .subspace U orthogonal basis for U dimension of UM ∩ M ∗ I M { A i } Di =0 D + 1 M ∗ { A ∗ i } Di =0 D + 1 M + M ∗ { A D , . . . , A , I, A ∗ , . . . , A ∗ D } D + 1 M M ∗ ∩ M ∗ M { A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } D + 1 + PM M ∗ { A i A ∗ j | ≤ i, j ≤ D } ( D + 1) M ∗ M { A ∗ j A i | ≤ i, j ≤ D } ( D + 1) M M ∗ + M ∗ M { A i A ∗ j + A ∗ j A i , A i A ∗ j − A ∗ j A i D + 2 D + 1 − P | ≤ i, j ≤ D, u i ( θ j ) = ± }∪{ A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } Theorem 37.
In each row of the table below we describe an edge U ⊆ W from thediagram of Figure 1. We give an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement. U W orthogonal basis dimensionfor U ⊥ ∩ W of U ⊥ ∩ WM ∩ M ∗ M { A i } Di =1 DM ∩ M ∗ M ∗ { A ∗ i } Di =1 DM M + M ∗ { A ∗ i } Di =1 DM ∗ M + M ∗ { A i } Di =1 DM + M ∗ M M ∗ ∩ M ∗ M { A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } PM M ∗ ∩ M ∗ M M M ∗ { A i A ∗ j | ≤ i, j ≤ D, u i ( θ j ) = ± } D − PM M ∗ ∩ M ∗ M M ∗ M { A ∗ j A i | ≤ i, j ≤ D, u i ( θ j ) = ± } D − PM M ∗ M M ∗ + M ∗ M { u i ( θ j ) A i A ∗ j − A ∗ j A i | ≤ i, j ≤ D, D − Pu i ( θ j ) = ± } M ∗ M M M ∗ + M ∗ M { A i A ∗ j − u i ( θ j ) A ∗ j A i | ≤ i, j ≤ D, D − Pu i ( θ j ) = ± } In this section, we give some open problems and suggestions for future research.Earlier in the paper we discussed the diagram of Figure 1. In this discussion weanalyzed the diagram up to
M M ∗ + M ∗ M . The remaining part of the diagram isnot completely understood. We mention what is known. By Lemma 1 the subspace M ∗ M M ∗ has an orthogonal basis { E ∗ i A j E ∗ h | ≤ h, i, j ≤ D, p hij = 0 } . Similarly, thesubspace M M ∗ M has an orthogonal basis { E i A ∗ j E h | ≤ h, i, j ≤ D, q hij = 0 } . Problem 38.
Find an orthogonal basis for the following subspaces:(i)
M M ∗ M ∩ M ∗ M M ∗ , (ii) M M ∗ M + M ∗ M M ∗ . Problem 39.
In each row of the table below we give an edge U ⊆ W from thediagram of Figure 1. Find an orthogonal basis for the orthogonal complement of U in W .4 U WM M ∗ + M ∗ M M M ∗ M ∩ M ∗ M M ∗ M M ∗ M ∩ M ∗ M M ∗ M M ∗ MM M ∗ M ∩ M ∗ M M ∗ M ∗ M M ∗ M M ∗ M M M ∗ M + M ∗ M M ∗ M ∗ M M ∗ M M ∗ M + M ∗ M M ∗ The author would like to thank Professor Paul Terwilliger for many valuable ideasand insightful suggestions on my work. This paper was written while the author wasan Honorary Fellow at the University of Wisconsin-Madison (January 2017 – Jan-uary 2018) supported by the Development and Promotion of Science and TechnologyTalents (DPST) Project, Thailand.
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