READING REPORT Symmetric Jordan Basis, Terwilliger Algebra of Binary - PowerPoint PPT Presentation

READING REPORT Symmetric Jordan Basis, Terwilliger Algebra of Binary Hamming Scheme and Gelfand-Tsetlin Basis Yizhe Zhu Shanghai Jiao Tong University zyzwstc@sjtu.edu.cn December 13, 2014 Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 1 / 41

Overview Symmetric Jordan Basis 1 a Linear Analog of SCD the Linear BTK Algorithm SJB of V ( B ( n )) Terwilliger algebra of binary Hamming scheme 2 Association Schemes Block Diagonalization Gelfand-Tsetlin basis 3 Branching Graph Gelfand-Tsetlin Basis Gelfand-Tsetlin Diagrams Remark 4 Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 2 / 41

Symmetric Chain Decomposition (SCD) Definition (SCD) Let P be a finite graded poset with n = r ( P ),where r : P → N is the rank function .If p ≤ q , we have r ( p ) ≤ r ( q ). We say p covers q in P if r ( p ) = r ( q ) + 1. The rank of P is r ( P ) = max { r ( p ) : p ∈ P } . Let P i denote the set of elements in P with rank i . A symmetric chain of a graded poset P is a sequence ( p 1 , ..., p h ) of elements in P such that p i covers p i − 1 for i = 2 , ..., h , and r ( p 1 ) + r ( p h ) = r ( P ), if h ≥ 2 or 2 r ( p 1 ) = r ( P ), if h = 1. A symmetric chain decomposition (SCD) of P is a decomposition of P into pairwise disjoint symmetric chains. Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 3 / 41

Symmetric Chain Decomposition (SCD) Example (SCD of B (4), Greene - Kleitman) Figure: the symmetric chain decomposition of B (4) Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 4 / 41

Symmetric Jordan Basis Definition (SJB) P is defined as before, let V ( P ) denote the complex vector space with P as basis. Then we have V ( P ) = V ( P 0 ) ⊕ V ( P 1 ) ⊕ ... ⊕ V ( P n ). An element v ∈ V ( P ) is homogeneous if v ∈ V ( P i ) for some i . A linear map U : V ( P ) → V ( P ) is said to be up operator if for all p ∈ P , U ( p ) = � q q is the sum of the all elements covering p . We define U ( p ) = 0 if p is a maximal element of P . A Jordan chain in V ( P ) is a sequence v = ( v 1 , ..., v h ) of nonzero homogeneous elements such that U ( v i − 1 ) = v i for i = 2 , ..., h and U ( v h ) = 0. We say v is symmetric , if r ( v 1 ) + r ( v h ) = r ( P ) for h ≥ 2 or 2 r ( v 1 ) = r ( P ) for h = 1. A symmetric Jordan basis (SJB) of V ( P ) is a basis of V ( P ) consisting of a disjoint union of symmetric Jordan chains in V ( P ). Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 5 / 41

Symmetric Jordan Basis Example The SJB of V ( B (2)) is given by two chains: ((0 , 0) , (1 , 0) + (0 , 1) , 2(1 , 1)), ((0 , 1) − (1 , 0)). Because r ((0 , 0)) + r (2(1 , 1)) = 2, 2 r ((0 , 1) − (1 , 0)) = 2, U ((0 , 0)) = (1 , 0) + (0 , 1), U ((1 , 0) + (0 , 1)) = (1 , 1) + (1 , 1) = 2(1 , 1) and U ((0 , 1) − (1 , 0)) = (0 , 0). Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 6 / 41

The Linear BTK Algorithm Let k 1 , ... k n be nonnegative integers. Define the poset M ( n , k 1 , ..., k n ) = { ( x 1 , ..., x n ) ∈ N n : 0 ≤ x i ≤ k i , for all i } with partial order defined by componentwise ≤ . When k 1 = k 2 = , , , = k n we write M ( n , k ) for M ( n , k ..., k ). Moreover, when k = 1, it is Boolean algebra B ( n ). An algorithm to construct an explicit SCD of M ( n , k 1 , ..., k n ) was given by de Bruijin, Tengbergen and Kruyswijk, called BTK. Here we present a linear analog of BTK algorithm. Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 7 / 41

The Linear BTK Algorithm The basic bulding block of the linear BTK is an inductive method for constructing a SJB of V ( M (2 , p , q )). Lemma Let p , q be positive and set P = M (2 , p , q ) , W = V ( P ) with up operator U. dimW = ( p + 1)( q + 1) . The action of U on the standard basis of W is given as follows: for 0 ≤ i ≤ p , 0 ≤ j ≤ q  ( i + 1 , j ) + ( i , j + 1) if i < p , j < q   ( i + 1 , j ) if i < p , j = q  U (( i , j )) = ( i , j + 1) if i = p , j < q   0 if i = p , j = q  Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 8 / 41

The Linear BTK Algorithm Let v = ( v (0) , v (1) , v (2) , ..., v ( p + q )) be the symmetric Jordan chain in � k � W generated by v (0) = (0 , 0). Then v ( k ) = � ( i , j ) , 0 ≤ k ≤ p + q . i , j i Define the homogeneous vector in W = V ( M (2 , p , q )) as follows. v ( i , j ) = ( p − i )( i , j ) − ( q − j + 1)( i + 1 , j − 1) , 0 ≤ i ≤ p − 1 , 1 ≤ j ≤ q . Theorem (1) { v ( k ) | o ≤ k ≤ p + q } ∪ { v ( i , j ) | 0 ≤ j ≤ p − 1 , 1 ≤ j ≤ q } is a basis of W . (2) For 0 ≤ i ≤ p − 1 , 1 ≤ j ≤ q we have  v ( i + 1 , j ) + v ( i , j + 1) if i < p − 1 , j < q   v ( i + 1 , j ) if i < p − 1 , j = q  U ( v ( i , j )) = v ( i , j + 1) if i = p − 1 , j < q   0 if i = p − 1 , j = q  Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 9 / 41

The Linear BTK Algorithm Thus the action of U on the v ( i , j ) is isomorphic to the action of U on the standard basis ( i , j ) of V ( M (2 , p − 1 , q − 1)), except that the map ( i , j ) → v ( i , j + 1) , ( i , j ) ∈ M (2 , p − q , q − 1) shifts ranks by one. We use the following example to show the induction works. Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 10 / 41

Example (SJB of V ( M (2 , 2 , 2))) The SJB of V ( M (2 , 1 , 1)) is given by two chains: ((0 , 0) , (1 , 0) + (0 , 1) , 2(1 , 1)) ((0 , 1) − (1 , 0)). Transferring these chains to V ( M (2 , 2 , 2)) via the map ( i , j ) → v ( i , j + 1) gives the following chains. ( v (0 , 1) , v (1 , 1) + v (0 , 2) , 2 v (1 , 2)) (I) ( v (0 , 2) − v (1 , 1)) (II) where v ( i , j ) are given by v (0 , 1) = 2(0 , 1) − 2(1 , 0) v (1 , 1) = (1 , 1) − 2(2 , 0) v (0 , 2) = 2(0 , 2) − (1 , 1) v (1 , 2) = (1 , 2) − (2 , 1) The symmetric Jordan chain generated by v (0) = (0 , 0) is given by ((0 , 0) , (1 , 0) + (0 , 1) , (2 , 0) + 2(1 , 1) + (0 , 2) , 3(2 , 1) + 3(1 , 2) , 6(2 , 2)) (III) Chains (I), (II), (III) form a SJB of V ( M (2 , 2 , 2)) Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 11 / 41

Induction Let BTK (2 , p , q ) be the set of SJB of V ( M (2 , p , q )) produced by the linear BTK algorithm. Now we can summarize the induction from BTK (2 , p − 1 , q − 1) to BTK (2 , p , q ) as follows. 1 For any ( i , j ) ∈ BTK (2 , p − 1 , q − 1), the map ( i , j ) → v ( i , j + 1) creates the elements in BTK (2 , p , q ) and keeps the symmetric Jordan chain. 2 Starting from v (0) = (0 , 0) ,the up operator U creates a symmetric Jordan chain v = ( v (0) , v (1) , v (2) , ..., v ( p + q )). 3 The symmetric Jordan chains created by Step 1 and Step 2 form the BTK (2 , p , q ). The case V = V ( M ( n , k 1 , ..., k n )) , n ≥ 3 can be reduced to the case n = 2 by induction. Then we complete the linear BTK algorithm. Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 12 / 41

SJB of V ( B ( n )) When applied to V ( B ( n )), the linear BTK algorithm can produce an SJB with very interesting properties. Let � , � denote the standard inner product on V ( B ( n )), i.e., � X , Y � = δ ( X , Y ), (Kronecker delta) for X , Y ∈ B ( n ). The length � � v , v � of v ∈ V ( B ( n )) is denoted � v � . Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 13 / 41

SJB of V ( B ( n )) Proposition Let O ( n ) be the SJB produced by linear BTK when appled to B ( n ). (1) The elements of O ( n ) are orthogonal wrt the standard inner product. (2) Let 0 ≤ k ≤ [ n / 2] and let ( x k , ..., x n − k ) and ( y k , ..., y n − k ) be any two symmetric Jordan chains in O ( n ) starting at rank k and ending at rank n − k . Then � x u +1 � � x u � = � y u +1 � � y u � , k ≤ u < n − k . (3) In the notation of (2), we have, for k ≤ u < u − k , � 1 2 � n − 2 k � − 1 � x u +1 � � n − 2 k � � x u � = ( u + 1 − k )( n − k − u ) = ( n − k − u ) 2 u − k u +1 − k Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 14 / 41

Example (SJB of V(B(n))) (1) The SJB of V ( B (2)) is given by ((0 , 0) , (1 , 0) + (0 , 1) , 2(1 , 1)) ((0 , 1) − (1 , 0)) (2) The SJB of V ( B (3)) is given by ((0 , 0 , 0) , (1 , 0 , 0) + (0 , 1 , 0) + (0 , 0 , 1) , 2((1 , 1 , 0) + (1 , 0 , 1) + (0 , 1 , 1)) , 6(1 , 1 , 1)) (2(0 , 0 , 1) − (1 , 0 , 0) − (0 , 1 , 0) , (1 , 0 , 1) + (0 , 1 , 1) − 2(1 , 1 , 0)) ((0 , 1 , 0) − (1 , 0 , 0) , (0 , 1 , 1) − (1 , 0 , 1)) Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 15 / 41

SJB of V ( B ( n )) More interesting properties can be said about SJB of V ( B ( n )) from Terwilliger algebra and Gelfand-Tsetlin basis. Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 16 / 41

Symmetric Association Schemes Definition (symmetric association schemes) A symmetric association scheme of class d is a pair Y = ( X , { R i } d i =0 ) consisting of a finite set X amd relations R 0 , R 1 , ..., R d on X such that: { R i } d i =0 is a partition of X × X . R 0 = { ( x , x ) : x ∈ X } . R i = R t i for 0 ≤ i ≤ d , where R t i = { ( y , x ) : ( x , y ) ∈ R i } . Given ( x , y ) ∈ R h , p h i , j = |{ z ∈ X : ( x , z ) ∈ R i , ( z , y ) ∈ R j }| depends only on h , i and j . Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 17 / 41