Tridiagonal pairs of shape (1, 2, 1)

• Main Author: Vidar, Melvin A.
• Format: text
• Language: English
• Published: Animo Repository 2008
• Subjects: Orthogonal polynomials; Shape; Transformations (Mathematics); Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/188; https://animorepository.dlsu.edu.ph/context/etd_doctoral/article/1187/viewcontent/CDTG004399_P.pdf
• Institution: De La Salle University

This dissertation is about tridiagonal pairs of shape (1, 2, 1). It is the simplest case of a family of tridiagonal pairs of shape (1, 2, 2, . . . , 2, 2, 1) which is related to P- and Qpolynomial association schemes. Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V ! V and A : V ! V that satisfies the following conditions: (i) each of A,A is diagonalizable (ii) there exists an ordering {Vi}d i=0 of the eigenspaces of A such that A Vi Vi1 + Vi + Vi+1 for 0 i d, where V1 = 0 and Vd+1 = 0 (iii) there exists an ordering {V i } i=0 of the eigenspaces of A such that AV i V i1+V i +V i+1 for 0 i , where V 1 = 0 and V +1 = 0 (iv) there is no subspace W of V such that AW W, A W W, W 6= 0,W 6= V . We call such a pair a tridiagonal pair on V . It is known that d = and that for 0 i d the dimensions of Vi, Vdi, V i , V di coincide we denote this common value by i. The sequence { i}d i=0 is called the shape of the pair. In this dissertation we assume the shape is (1, 2, 1) and obtain the following results. We describe six bases for V one diagonalizes A, another diagonalizes A , and the other four underlie the split decompositions for A,A . We give the action of A and A on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape (1, 2, 1) in terms of a sequence of scalars called the parameter array.