DTG → P -polynomial Association Schemes The equation D � p k A i A j = A j A i = ij A k k =0 shows that { A 0 , A 1 , . . . , A D } generate a matrix algebra over R (of dimension D + 1), called the Bose-Mesner algebra of Γ. Moreover, when AS comes from DTG, A i = v i ( A 1 ), for some polynomials v i , i = 0 , 1 , . . . , D , of degree i . This property is called P - polynomiality . DTG � P -polynomial AS
P -polynomial Association Schemes = DRG DTG � P -polynomial AS In a P -polynomial AS, for some ordering of A 0 , A 1 , . . . , A D : A i = v i ( A 1 ), for some polynomials v i , i = 0 , 1 , . . . , D , of degree i . A 1 can be considered as the adjacency matrix of a graph, while other A i ’s as its distance- i matrices. Such a graph is called Distance-Regular (DRG) . ∂ ( x, y ) = k ⇒ p k ij := |{ z ∈ V : ∂ ( x, z ) = i, ∂ ( y, z ) = j }| .
P -polynomial Association Schemes = DRG DTG � P -polynomial AS In a P -polynomial AS, for some ordering of A 0 , A 1 , . . . , A D : A i = v i ( A 1 ), for some polynomials v i , i = 0 , 1 , . . . , D , of degree i . A 1 can be considered as the adjacency matrix of a graph, while other A i ’s as its distance- i matrices. Such a graph is called Distance-Regular (DRG) . ∂ ( x, y ) = k ⇒ p k ij := |{ z ∈ V : ∂ ( x, z ) = i, ∂ ( y, z ) = j }| .
DTG � P -polynomial AS = DRG The smallest DRG that is not DTG: The Shrikhande graph. ◮ Its distance- i matrices form P -polynomial AS; ◮ However, its group of automorphisms is not Distance-Transitive.
Dual operation on the Bose-Mesner algebra Let { A 0 , A 1 , . . . , A D } be a symmetric association scheme. All A i ’s can be simultaneously diagonalised, so that the matrix algebra generated by A i ’s has the second basis: E 0 , E 1 , . . . , E D , where E j is the orthogonal projection onto a maximal common eigenspace of all { A i } D i =0 . D D � � θ ∗ A i = θ ik E k , E i = ik A k , k =0 k =0 D � p k A i A j = ij A k , E i E j = δ ij E i , k =0 D � q k A i ◦ A j = δ ij A i , E i ◦ E j = ij E k , k =0 p k q k ij (the Krein parameters ) ij
Dual operation on the Bose-Mesner algebra Let { A 0 , A 1 , . . . , A D } be a symmetric association scheme. All A i ’s can be simultaneously diagonalised, so that the matrix algebra generated by A i ’s has the second basis: E 0 , E 1 , . . . , E D , where E j is the orthogonal projection onto a maximal common eigenspace of all { A i } D i =0 . D D � � θ ∗ A i = θ ik E k , E i = ik A k , k =0 k =0 D � p k A i A j = ij A k , E i E j = δ ij E i , k =0 D � q k A i ◦ A j = δ ij A i , E i ◦ E j = ij E k , k =0 p k q k ij (the Krein parameters ) ij
Dual operation on the Bose-Mesner algebra Let { A 0 , A 1 , . . . , A D } be a symmetric association scheme. All A i ’s can be simultaneously diagonalised, so that the matrix algebra generated by A i ’s has the second basis: E 0 , E 1 , . . . , E D , where E j is the orthogonal projection onto a maximal common eigenspace of all { A i } D i =0 . D D � � θ ∗ A i = θ ik E k , E i = ik A k , k =0 k =0 D � p k A i A j = ij A k , E i E j = δ ij E i , k =0 D � q k A i ◦ A j = δ ij A i , E i ◦ E j = ij E k , k =0 p k q k ij (the Krein parameters ) ij
Dual operation on the Bose-Mesner algebra Let { A 0 , A 1 , . . . , A D } be a symmetric association scheme. All A i ’s can be simultaneously diagonalised, so that the matrix algebra generated by A i ’s has the second basis: E 0 , E 1 , . . . , E D , where E j is the orthogonal projection onto a maximal common eigenspace of all { A i } D i =0 . D D � � θ ∗ A i = θ ik E k , E i = ik A k , k =0 k =0 D � p k A i A j = ij A k , E i E j = δ ij E i , k =0 D � q k A i ◦ A j = δ ij A i , E i ◦ E j = ij E k , k =0 p k q k ij (the Krein parameters ) ij
Dual operation on the Bose-Mesner algebra Let { A 0 , A 1 , . . . , A D } be a symmetric association scheme. All A i ’s can be simultaneously diagonalised, so that the matrix algebra generated by A i ’s has the second basis: E 0 , E 1 , . . . , E D , where E j is the orthogonal projection onto a maximal common eigenspace of all { A i } D i =0 . D D � � θ ∗ A i = θ ik E k , E i = ik A k , k =0 k =0 D � p k A i A j = ij A k , E i E j = δ ij E i , k =0 D � q k A i ◦ A j = δ ij A i , E i ◦ E j = ij E k , k =0 p k q k ij (the Krein parameters ) ij
Duality and Q -polynomial Association Schemes Two bases of the Bose-Mesner algebra: { A 0 , A 1 , . . . , A D } { E 0 , E 1 , . . . , E D } For a P -polynomial association scheme: A i = v i ( A 1 ) Suppose now that, for some ordering of E j ’s: E j = v ∗ j ( E 1 ) , where v ∗ j is a ◦ -polynomial of degree j . This property is called Q - polynomiality , and it was introduced by P. Delsarte (1973) in his PhD Thesis ’ An Algebraic Approach to the Association Schemes of Coding Theory ’.
Duality and Q -polynomial Association Schemes Two bases of the Bose-Mesner algebra: { A 0 , A 1 , . . . , A D } { E 0 , E 1 , . . . , E D } For a P -polynomial association scheme: A i = v i ( A 1 ) Suppose now that, for some ordering of E j ’s: E j = v ∗ j ( E 1 ) , where v ∗ j is a ◦ -polynomial of degree j . This property is called Q - polynomiality , and it was introduced by P. Delsarte (1973) in his PhD Thesis ’ An Algebraic Approach to the Association Schemes of Coding Theory ’.
Duality and Q -polynomial Association Schemes Two bases of the Bose-Mesner algebra: { A 0 , A 1 , . . . , A D } { E 0 , E 1 , . . . , E D } For a P -polynomial association scheme: A i = v i ( A 1 ) Suppose now that, for some ordering of E j ’s: E j = v ∗ j ( E 1 ) , where v ∗ j is a ◦ -polynomial of degree j . This property is called Q - polynomiality , and it was introduced by P. Delsarte (1973) in his PhD Thesis ’ An Algebraic Approach to the Association Schemes of Coding Theory ’.
Duality and Q -polynomial Association Schemes Two bases of the Bose-Mesner algebra: { A 0 , A 1 , . . . , A D } { E 0 , E 1 , . . . , E D } For a P -polynomial association scheme: A i = v i ( A 1 ) Suppose now that, for some ordering of E j ’s: E j = v ∗ j ( E 1 ) , where v ∗ j is a ◦ -polynomial of degree j . This property is called Q - polynomiality , and it was introduced by P. Delsarte (1973) in his PhD Thesis ’ An Algebraic Approach to the Association Schemes of Coding Theory ’.
Error-Correcting Codes and Hamming Scheme Let X be an alphabet of m symbols, X n = X × X × . . . X ( n times) is the set of words of length n over X . The main idea of ECC is to increase the Hamming distance between words by mapping them from X k to X n , n > k . The Hamming graph H ( n, m ): ◮ the vertex set V = X n , | X | = m , ◮ the distance ∂ ( x, y ) is the Hamming distance, i.e., the number of different coordinates between words x and y . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a code is a subset of V (with large min distance ).
Error-Correcting Codes and Hamming Scheme Let X be an alphabet of m symbols, X n = X × X × . . . X ( n times) is the set of words of length n over X . The main idea of ECC is to increase the Hamming distance between words by mapping them from X k to X n , n > k . The Hamming graph H ( n, m ): ◮ the vertex set V = X n , | X | = m , ◮ the distance ∂ ( x, y ) is the Hamming distance, i.e., the number of different coordinates between words x and y . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a code is a subset of V (with large min distance ).
Error-Correcting Codes and Hamming Scheme Let X be an alphabet of m symbols, X n = X × X × . . . X ( n times) is the set of words of length n over X . The main idea of ECC is to increase the Hamming distance between words by mapping them from X k to X n , n > k . The Hamming graph H ( n, m ): ◮ the vertex set V = X n , | X | = m , ◮ the distance ∂ ( x, y ) is the Hamming distance, i.e., the number of different coordinates between words x and y . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a code is a subset of V (with large min distance ).
Error-Correcting Codes and Hamming Scheme Let X be an alphabet of m symbols, X n = X × X × . . . X ( n times) is the set of words of length n over X . The main idea of ECC is to increase the Hamming distance between words by mapping them from X k to X n , n > k . The Hamming graph H ( n, m ): ◮ the vertex set V = X n , | X | = m , ◮ the distance ∂ ( x, y ) is the Hamming distance, i.e., the number of different coordinates between words x and y . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a code is a subset of V (with large min distance ).
Designs and Johnson Scheme Let X be a set of v elements, � X � = { all k -element subsets of X } . k � X � A design D of strength t is a subset of such that: every k � X � element of appears in the same number of elements of D . t The Johnson graph J ( v, k ): � X � ◮ the vertex set V = , k ◮ the distance ∂ ( x, y ) is the Johnson metric, i.e., k − | x ∩ y | . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a design is a subset of V .
Designs and Johnson Scheme Let X be a set of v elements, � X � = { all k -element subsets of X } . k � X � A design D of strength t is a subset of such that: every k � X � element of appears in the same number of elements of D . t The Johnson graph J ( v, k ): � X � ◮ the vertex set V = , k ◮ the distance ∂ ( x, y ) is the Johnson metric, i.e., k − | x ∩ y | . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a design is a subset of V .
Designs and Johnson Scheme Let X be a set of v elements, � X � = { all k -element subsets of X } . k � X � A design D of strength t is a subset of such that: every k � X � element of appears in the same number of elements of D . t The Johnson graph J ( v, k ): � X � ◮ the vertex set V = , k ◮ the distance ∂ ( x, y ) is the Johnson metric, i.e., k − | x ∩ y | . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a design is a subset of V .
Designs and Johnson Scheme Let X be a set of v elements, � X � = { all k -element subsets of X } . k � X � A design D of strength t is a subset of such that: every k � X � element of appears in the same number of elements of D . t The Johnson graph J ( v, k ): � X � ◮ the vertex set V = , k ◮ the distance ∂ ( x, y ) is the Johnson metric, i.e., k − | x ∩ y | . ◮ distance-transitive and Q -polynomial ⇒ ( P and Q ), ◮ a design is a subset of V .
Delsarte’s Theory The main problem of coding theory (sphere-packing problem): ◮ maximize | C | subject to fixed min distance, ◮ maximize min distance subject to | C | . There are some fundamental bounds on | C | in terms of min distance and some other parameters. There are very similar bounds on |D| for a design D in terms of its strength t and some other parameters. Delsarte reproved all these bounds in a uniform way. To do so, he realized that codes and designs should be considered as subsets of vertices of certain ( P and Q )-polynomial scheme. Roughly speaking, deriving those bounds in terms of: ◮ ’distance’ requires P -polynomiality (we need a metric), ◮ ’strength’ requires Q -polynomiality ( ∗ -metric).
Delsarte’s Theory The main problem of coding theory (sphere-packing problem): ◮ maximize | C | subject to fixed min distance, ◮ maximize min distance subject to | C | . There are some fundamental bounds on | C | in terms of min distance and some other parameters. There are very similar bounds on |D| for a design D in terms of its strength t and some other parameters. Delsarte reproved all these bounds in a uniform way. To do so, he realized that codes and designs should be considered as subsets of vertices of certain ( P and Q )-polynomial scheme. Roughly speaking, deriving those bounds in terms of: ◮ ’distance’ requires P -polynomiality (we need a metric), ◮ ’strength’ requires Q -polynomiality ( ∗ -metric).
Delsarte’s Theory The main problem of coding theory (sphere-packing problem): ◮ maximize | C | subject to fixed min distance, ◮ maximize min distance subject to | C | . There are some fundamental bounds on | C | in terms of min distance and some other parameters. There are very similar bounds on |D| for a design D in terms of its strength t and some other parameters. Delsarte reproved all these bounds in a uniform way. To do so, he realized that codes and designs should be considered as subsets of vertices of certain ( P and Q )-polynomial scheme. Roughly speaking, deriving those bounds in terms of: ◮ ’distance’ requires P -polynomiality (we need a metric), ◮ ’strength’ requires Q -polynomiality ( ∗ -metric).
Bannai’s Observation A spherical t - design : points x 1 , x 2 , . . . , x N ∈ S d such that � N � S d f ( x ) dµ d ( x ) = 1 f ( x i ) N i =1 for all polynomials f in d + 1 variables, of total degree ≤ t . It turned out that some bounds for t -designs, whose proofs required Q -polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q -polynomial AS P -polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space = by H.C. Wang (1952) � �� � Classified by E. Cartan (1926)
Bannai’s Observation A spherical t - design : points x 1 , x 2 , . . . , x N ∈ S d such that � N � S d f ( x ) dµ d ( x ) = 1 f ( x i ) N i =1 for all polynomials f in d + 1 variables, of total degree ≤ t . It turned out that some bounds for t -designs, whose proofs required Q -polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q -polynomial AS P -polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space = by H.C. Wang (1952) � �� � Classified by E. Cartan (1926)
Bannai’s Observation A spherical t - design : points x 1 , x 2 , . . . , x N ∈ S d such that � N � S d f ( x ) dµ d ( x ) = 1 f ( x i ) N i =1 for all polynomials f in d + 1 variables, of total degree ≤ t . It turned out that some bounds for t -designs, whose proofs required Q -polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q -polynomial AS P -polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space = by H.C. Wang (1952) � �� � Classified by E. Cartan (1926)
Bannai’s Observation A spherical t - design : points x 1 , x 2 , . . . , x N ∈ S d such that � N � S d f ( x ) dµ d ( x ) = 1 f ( x i ) N i =1 for all polynomials f in d + 1 variables, of total degree ≤ t . It turned out that some bounds for t -designs, whose proofs required Q -polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q -polynomial AS P -polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space = by H.C. Wang (1952) � �� � Classified by E. Cartan (1926)
Bannai’s Observation A spherical t - design : points x 1 , x 2 , . . . , x N ∈ S d such that � N � S d f ( x ) dµ d ( x ) = 1 f ( x i ) N i =1 for all polynomials f in d + 1 variables, of total degree ≤ t . It turned out that some bounds for t -designs, whose proofs required Q -polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. =? Q -polynomial AS P -polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space = by H.C. Wang (1952) � �� � Classified by E. Cartan (1926)
Bannai’s Conjecture (early 1980s) (1) If diameter D is large enough, then primitive P -polynomial association scheme is Q -polynomial, and vice versa. Seems to be beyond the scope of our reach at this moment. (2) All ( P and Q )-polynomial association schemes are known. Bannai made a list of ( P and Q )-polynomial association schemes as a finite-analogue of Cartan’s classification.
Bannai’s Conjecture (early 1980s) (1) If diameter D is large enough, then primitive P -polynomial association scheme is Q -polynomial, and vice versa. Seems to be beyond the scope of our reach at this moment. (2) All ( P and Q )-polynomial association schemes are known. Bannai made a list of ( P and Q )-polynomial association schemes as a finite-analogue of Cartan’s classification.
Bannai’s Conjecture (early 1980s) (1) If diameter D is large enough, then primitive P -polynomial association scheme is Q -polynomial, and vice versa. Seems to be beyond the scope of our reach at this moment. (2) All ( P and Q )-polynomial association schemes are known. Bannai made a list of ( P and Q )-polynomial association schemes as a finite-analogue of Cartan’s classification.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
(known) ( P and Q )-Polynomial Association Schemes Main families (except of schemes of small diameter): ◮ Schemes of dual polar spaces ◮ Vertices: maximal isotropic subspaces of a vector space equipped with a form (symplectic/quadratic/Hermitian). ◮ Distance ∂ : codimension of their intersection. ◮ Schemes of sesquilinear/quadratic forms ◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂ : rank of their difference. ◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen (2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a ( P and Q )-polynomial AS from the above.
Classification Conjecture (2) All ( P and Q )-polynomial association schemes are known. The solution requires two steps: (A) To find all feasible parameters of ( P and Q )-polynomial AS. ( P and Q )-polynomiality of an AS is determined by its intersection numbers { p k ij } . We want to describe all feasible sets { p k ij } corresponding to ( P and Q )-polynomial AS. (In progress. Most of the work done by D. Leonard, P. Terwilliger, T. Ito and their school.) (B) To characterise ( P and Q )-polynomial AS by parameters. There may exist two or more schemes with the same parameters. We want to find all of them up to isomorphism. (Not much progress since 1999.)
Classification Conjecture (2) All ( P and Q )-polynomial association schemes are known. The solution requires two steps: (A) To find all feasible parameters of ( P and Q )-polynomial AS. ( P and Q )-polynomiality of an AS is determined by its intersection numbers { p k ij } . We want to describe all feasible sets { p k ij } corresponding to ( P and Q )-polynomial AS. (In progress. Most of the work done by D. Leonard, P. Terwilliger, T. Ito and their school.) (B) To characterise ( P and Q )-polynomial AS by parameters. There may exist two or more schemes with the same parameters. We want to find all of them up to isomorphism. (Not much progress since 1999.)
Classification Conjecture (2) All ( P and Q )-polynomial association schemes are known. The solution requires two steps: (A) To find all feasible parameters of ( P and Q )-polynomial AS. ( P and Q )-polynomiality of an AS is determined by its intersection numbers { p k ij } . We want to describe all feasible sets { p k ij } corresponding to ( P and Q )-polynomial AS. (In progress. Most of the work done by D. Leonard, P. Terwilliger, T. Ito and their school.) (B) To characterise ( P and Q )-polynomial AS by parameters. There may exist two or more schemes with the same parameters. We want to find all of them up to isomorphism. (Not much progress since 1999.)
The Grassmann graph J q ( n, d ) ◮ Let q ≥ 2 be a prime power, n ≥ d ≥ 1 be integers. ◮ J q ( n, d ) has as vertices all d -dim. subspaces U ≤ F n q . ◮ ∂ ( U 1 , U 2 ) = d − dim ( U 1 ∩ U 2 ). ◮ All intersection numbers p k ij are expressed in q, n, d . ◮ J q ( n, d ) ∼ = J q ( n, n − d ), diameter equals min( d, n − d ). Theorem (K. Metsch, 1995) The Grassmann graph J q ( n, d ), d > 2, is characterized by its intersection array with the following possible exceptions : ◮ n = 2 d , n = 2 d ± 1, ◮ n = 2 d ± 2 if q ∈ { 2 , 3 } , ◮ n = 2 d ± 3 if q = 2. A real exception, the twisted Grassmann graph with the same parameters as J q (2 d ± 1 , d ) for all q , was constructed by E.R. van Dam and J.H. Koolen, Invent. Math. (2005).
The Grassmann graph J q ( n, d ) ◮ Let q ≥ 2 be a prime power, n ≥ d ≥ 1 be integers. ◮ J q ( n, d ) has as vertices all d -dim. subspaces U ≤ F n q . ◮ ∂ ( U 1 , U 2 ) = d − dim ( U 1 ∩ U 2 ). ◮ All intersection numbers p k ij are expressed in q, n, d . ◮ J q ( n, d ) ∼ = J q ( n, n − d ), diameter equals min( d, n − d ). Theorem (K. Metsch, 1995) The Grassmann graph J q ( n, d ), d > 2, is characterized by its intersection array with the following possible exceptions : ◮ n = 2 d , n = 2 d ± 1, ◮ n = 2 d ± 2 if q ∈ { 2 , 3 } , ◮ n = 2 d ± 3 if q = 2. A real exception, the twisted Grassmann graph with the same parameters as J q (2 d ± 1 , d ) for all q , was constructed by E.R. van Dam and J.H. Koolen, Invent. Math. (2005).
The Grassmann graph J q ( n, d ) ◮ Let q ≥ 2 be a prime power, n ≥ d ≥ 1 be integers. ◮ J q ( n, d ) has as vertices all d -dim. subspaces U ≤ F n q . ◮ ∂ ( U 1 , U 2 ) = d − dim ( U 1 ∩ U 2 ). ◮ All intersection numbers p k ij are expressed in q, n, d . ◮ J q ( n, d ) ∼ = J q ( n, n − d ), diameter equals min( d, n − d ). Theorem (K. Metsch, 1995) The Grassmann graph J q ( n, d ), d > 2, is characterized by its intersection array with the following possible exceptions : ◮ n = 2 d , n = 2 d ± 1, ◮ n = 2 d ± 2 if q ∈ { 2 , 3 } , ◮ n = 2 d ± 3 if q = 2. A real exception, the twisted Grassmann graph with the same parameters as J q (2 d ± 1 , d ) for all q , was constructed by E.R. van Dam and J.H. Koolen, Invent. Math. (2005).
The bilinear forms graphs Bil q ( n × m ) ◮ Let q ≥ 2 be a prime power, n ≥ m ≥ 1 be integers. ◮ Bil q ( n × m ) has as vertices all n × m -matrices over F q . ◮ ∂ ( A, B ) = rank ( A − B ). ◮ All intersection numbers p k ij are expressed in q, n, m . ◮ Bil q ( n × m ) ∼ = Bil q ( m × n ), diameter equals min( n, m ). Theorem (K. Metsch, 1999) The bilinear forms graph Bil q ( n × m ), n ≥ m ≥ 3, is characterized by its intersection array with the following possible exceptions : ◮ q = 2 and m ∈ { n, n + 1 , n + 2 , n + 3 } , ◮ q ≥ 3 and m ∈ { n, n + 1 , n + 2 } . No actual exceptions are known.
The bilinear forms graphs Bil q ( n × m ) ◮ Let q ≥ 2 be a prime power, n ≥ m ≥ 1 be integers. ◮ Bil q ( n × m ) has as vertices all n × m -matrices over F q . ◮ ∂ ( A, B ) = rank ( A − B ). ◮ All intersection numbers p k ij are expressed in q, n, m . ◮ Bil q ( n × m ) ∼ = Bil q ( m × n ), diameter equals min( n, m ). Theorem (K. Metsch, 1999) The bilinear forms graph Bil q ( n × m ), n ≥ m ≥ 3, is characterized by its intersection array with the following possible exceptions : ◮ q = 2 and m ∈ { n, n + 1 , n + 2 , n + 3 } , ◮ q ≥ 3 and m ∈ { n, n + 1 , n + 2 } . No actual exceptions are known.
q -analogues of Hamming and Johnson schemes The Hamming and the Johnson schemes play an important roles in the classical coding and design theories, serving as underlying spaces for codes and designs. The Grassmann scheme and the bilinear forms scheme play the same roles for a new branch of coding theory: Subspace (or Network) Coding. ◮ a finite set → a vector space over F q , ◮ a codeword over an alphabet → a subspace.
q -analogues of Hamming and Johnson schemes The Hamming and the Johnson schemes play an important roles in the classical coding and design theories, serving as underlying spaces for codes and designs. The Grassmann scheme and the bilinear forms scheme play the same roles for a new branch of coding theory: Subspace (or Network) Coding. ◮ a finite set → a vector space over F q , ◮ a codeword over an alphabet → a subspace.
q -analogues of Hamming and Johnson schemes The Hamming and the Johnson schemes play an important roles in the classical coding and design theories, serving as underlying spaces for codes and designs. The Grassmann scheme and the bilinear forms scheme play the same roles for a new branch of coding theory: Subspace (or Network) Coding. ◮ a finite set → a vector space over F q , ◮ a codeword over an alphabet → a subspace.
Our results We showed that the following possible exceptions from the Metsch results cannot be realized : The Grassmann graph J q ( n, d ): q Possible exceptions by Metsch Ruled out (G., Koolen) ∀ q n = 2 d, n = 2 d ± 1 q = 2 , n = 2 d 2 , 3 n = 2 d ± 2 q = 2 , d is odd and ≫ 0 2 n = 2 d ± 3 d ≡ 1(mod 3) (in preparation) The bilinear forms graphs Bil q ( n × m ): Possible exceptions by Metsch Ruled out (G., Koolen) q 2 m ∈ { n, n + 1 , n + 2 , n + 3 } n = m ≥ 3 m ∈ { n, n + 1 , n + 2 } ( Combinatorica , to appear)
Our results We showed that the following possible exceptions from the Metsch results cannot be realized : The Grassmann graph J q ( n, d ): q Possible exceptions by Metsch Ruled out (G., Koolen) ∀ q n = 2 d, n = 2 d ± 1 q = 2 , n = 2 d 2 , 3 n = 2 d ± 2 q = 2 , d is odd and ≫ 0 2 n = 2 d ± 3 d ≡ 1(mod 3) (in preparation) The bilinear forms graphs Bil q ( n × m ): Possible exceptions by Metsch Ruled out (G., Koolen) q 2 m ∈ { n, n + 1 , n + 2 , n + 3 } n = m ≥ 3 m ∈ { n, n + 1 , n + 2 } ( Combinatorica , to appear)
Our results We showed that the following possible exceptions from the Metsch results cannot be realized : The Grassmann graph J q ( n, d ): q Possible exceptions by Metsch Ruled out (G., Koolen) ∀ q n = 2 d, n = 2 d ± 1 q = 2 , n = 2 d 2 , 3 n = 2 d ± 2 q = 2 , d is odd and ≫ 0 2 n = 2 d ± 3 d ≡ 1(mod 3) (in preparation) The bilinear forms graphs Bil q ( n × m ): Possible exceptions by Metsch Ruled out (G., Koolen) q 2 m ∈ { n, n + 1 , n + 2 , n + 3 } n = m ≥ 3 m ∈ { n, n + 1 , n + 2 } ( Combinatorica , to appear)
Metsch’s approach Partial linear space is a set P of points and a set L of lines (subsets of P ): ◮ any line contains at least two points, ◮ any two points are on at most one line. Both J q ( n, d ) and Bil q ( n × m ) naturally give rise to partial linear spaces (well studied in finite geometry): ◮ points = vertices, ◮ lines = maximal cliques. Let X denote J q ( n, d ) or Bil q ( n × m ). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X .
Metsch’s approach Partial linear space is a set P of points and a set L of lines (subsets of P ): ◮ any line contains at least two points, ◮ any two points are on at most one line. Both J q ( n, d ) and Bil q ( n × m ) naturally give rise to partial linear spaces (well studied in finite geometry): ◮ points = vertices, ◮ lines = maximal cliques. Let X denote J q ( n, d ) or Bil q ( n × m ). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X .
Metsch’s approach Partial linear space is a set P of points and a set L of lines (subsets of P ): ◮ any line contains at least two points, ◮ any two points are on at most one line. Both J q ( n, d ) and Bil q ( n × m ) naturally give rise to partial linear spaces (well studied in finite geometry): ◮ points = vertices, ◮ lines = maximal cliques. Let X denote J q ( n, d ) or Bil q ( n × m ). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X .
Metsch’s approach Partial linear space is a set P of points and a set L of lines (subsets of P ): ◮ any line contains at least two points, ◮ any two points are on at most one line. Both J q ( n, d ) and Bil q ( n × m ) naturally give rise to partial linear spaces (well studied in finite geometry): ◮ points = vertices, ◮ lines = maximal cliques. Let X denote J q ( n, d ) or Bil q ( n × m ). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X .
Metsch’s approach Partial linear space is a set P of points and a set L of lines (subsets of P ): ◮ any line contains at least two points, ◮ any two points are on at most one line. Both J q ( n, d ) and Bil q ( n × m ) naturally give rise to partial linear spaces (well studied in finite geometry): ◮ points = vertices, ◮ lines = maximal cliques. Let X denote J q ( n, d ) or Bil q ( n × m ). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X .
Metsch’s approach Partial linear space is a set P of points and a set L of lines (subsets of P ): ◮ any line contains at least two points, ◮ any two points are on at most one line. Both J q ( n, d ) and Bil q ( n × m ) naturally give rise to partial linear spaces (well studied in finite geometry): ◮ points = vertices, ◮ lines = maximal cliques. Let X denote J q ( n, d ) or Bil q ( n × m ). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X .
Metsch’s approach Metsch developed a purely combinatorial method, which allows to show the existence of ’large’ cliques, if certain inequalities on p k ij hold, and to construct a partial linear space. However, in case of possible exceptions , these inequalities do not hold, so that the first crucial step fails: A graph Γ with the same parameters as X �↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X . And this is not an arithmetic problem: a partial linear space cannot be recovered from the twisted Grassmann graph.
Metsch’s approach Metsch developed a purely combinatorial method, which allows to show the existence of ’large’ cliques, if certain inequalities on p k ij hold, and to construct a partial linear space. However, in case of possible exceptions , these inequalities do not hold, so that the first crucial step fails: A graph Γ with the same parameters as X �↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X . And this is not an arithmetic problem: a partial linear space cannot be recovered from the twisted Grassmann graph.
Metsch’s approach Metsch developed a purely combinatorial method, which allows to show the existence of ’large’ cliques, if certain inequalities on p k ij hold, and to construct a partial linear space. However, in case of possible exceptions , these inequalities do not hold, so that the first crucial step fails: A graph Γ with the same parameters as X �↓ (from parameters) A partial linear space on cliques → ? Known partial linear space ↓ Γ ∼ = X . And this is not an arithmetic problem: a partial linear space cannot be recovered from the twisted Grassmann graph.
Our approach ◮ Q -polynomiality → the local structure The local graph at vertex x : Γ 1 ( x ) := { y ∈ V (Γ) : ∂ ( x, y ) = 1 } . The µ - graph of vertices x, y with ∂ ( x, y ) = 2: Γ 1 ( x, y ) := { z ∈ V (Γ) : ∂ ( x, z ) = 1 & ∂ ( z, y ) = 1 } . Using Q -polynomiality, we can restrict: ◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ -graphs. ◮ Reconstructing the local structure The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix. ◮ Reconstructing a graph by its local structure If we know Γ 1 ( x ), Γ 1 ( x, y ) for all x, y → ? Γ
Our approach ◮ Q -polynomiality → the local structure The local graph at vertex x : Γ 1 ( x ) := { y ∈ V (Γ) : ∂ ( x, y ) = 1 } . The µ - graph of vertices x, y with ∂ ( x, y ) = 2: Γ 1 ( x, y ) := { z ∈ V (Γ) : ∂ ( x, z ) = 1 & ∂ ( z, y ) = 1 } . Using Q -polynomiality, we can restrict: ◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ -graphs. ◮ Reconstructing the local structure The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix. ◮ Reconstructing a graph by its local structure If we know Γ 1 ( x ), Γ 1 ( x, y ) for all x, y → ? Γ
Our approach ◮ Q -polynomiality → the local structure The local graph at vertex x : Γ 1 ( x ) := { y ∈ V (Γ) : ∂ ( x, y ) = 1 } . The µ - graph of vertices x, y with ∂ ( x, y ) = 2: Γ 1 ( x, y ) := { z ∈ V (Γ) : ∂ ( x, z ) = 1 & ∂ ( z, y ) = 1 } . Using Q -polynomiality, we can restrict: ◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ -graphs. ◮ Reconstructing the local structure The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix. ◮ Reconstructing a graph by its local structure If we know Γ 1 ( x ), Γ 1 ( x, y ) for all x, y → ? Γ
Our approach ◮ Q -polynomiality → the local structure The local graph at vertex x : Γ 1 ( x ) := { y ∈ V (Γ) : ∂ ( x, y ) = 1 } . The µ - graph of vertices x, y with ∂ ( x, y ) = 2: Γ 1 ( x, y ) := { z ∈ V (Γ) : ∂ ( x, z ) = 1 & ∂ ( z, y ) = 1 } . Using Q -polynomiality, we can restrict: ◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ -graphs. ◮ Reconstructing the local structure The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix. ◮ Reconstructing a graph by its local structure If we know Γ 1 ( x ), Γ 1 ( x, y ) for all x, y → ? Γ
Our approach ◮ Q -polynomiality → the local structure The local graph at vertex x : Γ 1 ( x ) := { y ∈ V (Γ) : ∂ ( x, y ) = 1 } . The µ - graph of vertices x, y with ∂ ( x, y ) = 2: Γ 1 ( x, y ) := { z ∈ V (Γ) : ∂ ( x, z ) = 1 & ∂ ( z, y ) = 1 } . Using Q -polynomiality, we can restrict: ◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ -graphs. ◮ Reconstructing the local structure The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix. ◮ Reconstructing a graph by its local structure If we know Γ 1 ( x ), Γ 1 ( x, y ) for all x, y → ? Γ
Our approach ◮ Q -polynomiality → the local structure The local graph at vertex x : Γ 1 ( x ) := { y ∈ V (Γ) : ∂ ( x, y ) = 1 } . The µ - graph of vertices x, y with ∂ ( x, y ) = 2: Γ 1 ( x, y ) := { z ∈ V (Γ) : ∂ ( x, z ) = 1 & ∂ ( z, y ) = 1 } . Using Q -polynomiality, we can restrict: ◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ -graphs. ◮ Reconstructing the local structure The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix. ◮ Reconstructing a graph by its local structure If we know Γ 1 ( x ), Γ 1 ( x, y ) for all x, y → ? Γ
The Bose-Mesner algebra and intersection numbers The equation D � p k A i A j = A j A i = ij A k k =0 shows that { A 0 , A 1 , . . . , A D } generate a matrix algebra over R (of dimension D + 1), called the Bose-Mesner algebra of Γ. The numbers p k ij have simple combinatorial interpretation: ∂ ( x, y ) = k ⇒ p k ij := |{ z ∈ V : ∂ ( x, z ) = i, ∂ ( y, z ) = j }| .
Triple intersection numbers Let Γ be a Q -polynomial DRG with diameter D ≥ 3. Fix a triple of vertices x, y, z such that ∂ ( x, y ) = ∂ ( x, z ) = 1. Denote a triple intersection number [ ℓ, m, n ]: [ ℓ, m, n ] x,y,z = |{ w : ∂ ( x, w ) = ℓ, ∂ ( y, w ) = m, ∂ ( x, w ) = n }| . For example, Terwilliger (1995) showed that for i ≥ 2 [ i, i ± 1 , i ± 1] = κ i,δ [1 , 1 , 1] + τ i where δ = d ( y, z ) ∈ { 1 , 2 } , and κ i,δ and τ i are real scalars that do not depend on the choice of x, y, z . y x z
Triple intersection numbers Let Γ be a Q -polynomial DRG with diameter D ≥ 3. Fix a triple of vertices x, y, z such that ∂ ( x, y ) = ∂ ( x, z ) = 1. Denote a triple intersection number [ ℓ, m, n ]: [ ℓ, m, n ] x,y,z = |{ w : ∂ ( x, w ) = ℓ, ∂ ( y, w ) = m, ∂ ( x, w ) = n }| . For example, Terwilliger (1995) showed that for i ≥ 2 [ i, i ± 1 , i ± 1] = κ i,δ [1 , 1 , 1] + τ i where δ = d ( y, z ) ∈ { 1 , 2 } , and κ i,δ and τ i are real scalars that do not depend on the choice of x, y, z . y x z
Triple intersection numbers Let Γ be a Q -polynomial DRG with diameter D ≥ 3. Fix a triple of vertices x, y, z such that ∂ ( x, y ) = ∂ ( x, z ) = 1. Denote a triple intersection number [ ℓ, m, n ]: [ ℓ, m, n ] x,y,z = |{ w : ∂ ( x, w ) = ℓ, ∂ ( y, w ) = m, ∂ ( x, w ) = n }| . For example, Terwilliger (1995) showed that for i ≥ 2 [ i, i ± 1 , i ± 1] = κ i,δ [1 , 1 , 1] + τ i where δ = d ( y, z ) ∈ { 1 , 2 } , and κ i,δ and τ i are real scalars that do not depend on the choice of x, y, z . y x z
Triple intersection numbers → Terwilliger algebra Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D , denote by E ∗ i := E ∗ i ( x ) a diagonal matrix with rows and columns indexed by V (Γ), and defined by � 1 if ∂ ( x, y ) = i, ( E ∗ i ) y,y := 0 if ∂ ( x, y ) � = i. The dual Bose-Mesner algebra (w.r.t. x ) M ∗ := M ∗ ( x ) = span { E ∗ 0 , E ∗ 1 , . . . , E ∗ D } . The Terwilliger (or subconstituent ) algebra (w.r.t. x ) T := T ( x ) = �M , M ∗ � , where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that R V decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.
Triple intersection numbers → Terwilliger algebra Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D , denote by E ∗ i := E ∗ i ( x ) a diagonal matrix with rows and columns indexed by V (Γ), and defined by � 1 if ∂ ( x, y ) = i, ( E ∗ i ) y,y := 0 if ∂ ( x, y ) � = i. The dual Bose-Mesner algebra (w.r.t. x ) M ∗ := M ∗ ( x ) = span { E ∗ 0 , E ∗ 1 , . . . , E ∗ D } . The Terwilliger (or subconstituent ) algebra (w.r.t. x ) T := T ( x ) = �M , M ∗ � , where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that R V decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.
Triple intersection numbers → Terwilliger algebra Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D , denote by E ∗ i := E ∗ i ( x ) a diagonal matrix with rows and columns indexed by V (Γ), and defined by � 1 if ∂ ( x, y ) = i, ( E ∗ i ) y,y := 0 if ∂ ( x, y ) � = i. The dual Bose-Mesner algebra (w.r.t. x ) M ∗ := M ∗ ( x ) = span { E ∗ 0 , E ∗ 1 , . . . , E ∗ D } . The Terwilliger (or subconstituent ) algebra (w.r.t. x ) T := T ( x ) = �M , M ∗ � , where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that R V decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.
Triple intersection numbers → Terwilliger algebra Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D , denote by E ∗ i := E ∗ i ( x ) a diagonal matrix with rows and columns indexed by V (Γ), and defined by � 1 if ∂ ( x, y ) = i, ( E ∗ i ) y,y := 0 if ∂ ( x, y ) � = i. The dual Bose-Mesner algebra (w.r.t. x ) M ∗ := M ∗ ( x ) = span { E ∗ 0 , E ∗ 1 , . . . , E ∗ D } . The Terwilliger (or subconstituent ) algebra (w.r.t. x ) T := T ( x ) = �M , M ∗ � , where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that R V decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.
Terwilliger algebra Denote � 1 and � A := E ∗ 1 A 1 E ∗ J := E ∗ 1 JE ∗ 1 . One can see � N � 0 � A = . 0 0 where N — the adjacency matrix of Γ 1 ( x ). 1 ) y,z and [1 , 1 , 1] = ( � Note [ ℓ, m, n ] x,y,z = ( E ∗ 1 A m E ∗ ℓ A n E ∗ A ) 2 y,z Then [ i, i ± 1 , i ± 1] = κ i,δ [1 , 1 , 1] + τ i , imply that E ∗ 1 A i − 1 E ∗ i A i − 1 E ∗ 1 and E ∗ 1 A i E ∗ i − 1 A i E ∗ 1 are the polynomials (of degree 2) in � A and � J := E ∗ 1 JE ∗ 1 .
Terwilliger algebra Denote � 1 and � A := E ∗ 1 A 1 E ∗ J := E ∗ 1 JE ∗ 1 . One can see � N � 0 � A = . 0 0 where N — the adjacency matrix of Γ 1 ( x ). 1 ) y,z and [1 , 1 , 1] = ( � Note [ ℓ, m, n ] x,y,z = ( E ∗ 1 A m E ∗ ℓ A n E ∗ A ) 2 y,z Then [ i, i ± 1 , i ± 1] = κ i,δ [1 , 1 , 1] + τ i , imply that E ∗ 1 A i − 1 E ∗ i A i − 1 E ∗ 1 and E ∗ 1 A i E ∗ i − 1 A i E ∗ 1 are the polynomials (of degree 2) in � A and � J := E ∗ 1 JE ∗ 1 .
Terwilliger algebra Denote � 1 and � A := E ∗ 1 A 1 E ∗ J := E ∗ 1 JE ∗ 1 . One can see � N � 0 � A = . 0 0 where N — the adjacency matrix of Γ 1 ( x ). 1 ) y,z and [1 , 1 , 1] = ( � Note [ ℓ, m, n ] x,y,z = ( E ∗ 1 A m E ∗ ℓ A n E ∗ A ) 2 y,z Then [ i, i ± 1 , i ± 1] = κ i,δ [1 , 1 , 1] + τ i , imply that E ∗ 1 A i − 1 E ∗ i A i − 1 E ∗ 1 and E ∗ 1 A i E ∗ i − 1 A i E ∗ 1 are the polynomials (of degree 2) in � A and � J := E ∗ 1 JE ∗ 1 .
The Terwilliger polynomial of a Q -DRG E ∗ 1 A i − 1 E ∗ i A i − 1 E ∗ 1 and E ∗ 1 A i E ∗ i − 1 A i E ∗ 1 are the polynomials (of degree 2) in � A and � J := E ∗ 1 JE ∗ 1 . ◮ Terwilliger (early 1990’s): There exists a polynomial p T of degree 4 such that, for any vertex x ∈ Γ, and any non-principal eigenvalue η of Γ 1 ( x ) we have p T ( η ) ≥ 0. ◮ p T only depends on the intersection numbers of Γ and the Q -polynomial ordering of primitive idempotents of its Bose-Mesner algebra. ◮ We call p T the Terwilliger polynomial . References: • P. Terwilliger, Lecture Note on Terwilliger algebra (edited by H. Suzuki), 1993. • A.L.G., J.H. Koolen, The Terwilliger polynomial of a Q -polynomial distance-regular graph and its application to pseudo-partition graphs // Linear Algebra and Its Appl. (2015).
The Terwilliger polynomial of a Q -DRG E ∗ 1 A i − 1 E ∗ i A i − 1 E ∗ 1 and E ∗ 1 A i E ∗ i − 1 A i E ∗ 1 are the polynomials (of degree 2) in � A and � J := E ∗ 1 JE ∗ 1 . ◮ Terwilliger (early 1990’s): There exists a polynomial p T of degree 4 such that, for any vertex x ∈ Γ, and any non-principal eigenvalue η of Γ 1 ( x ) we have p T ( η ) ≥ 0. ◮ p T only depends on the intersection numbers of Γ and the Q -polynomial ordering of primitive idempotents of its Bose-Mesner algebra. ◮ We call p T the Terwilliger polynomial . References: • P. Terwilliger, Lecture Note on Terwilliger algebra (edited by H. Suzuki), 1993. • A.L.G., J.H. Koolen, The Terwilliger polynomial of a Q -polynomial distance-regular graph and its application to pseudo-partition graphs // Linear Algebra and Its Appl. (2015).
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