Tridiagonal pairs of Krawtchouk type

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Let F denote an algebraically closed field with characteristic 0 and let V denote a vector space over F with finite positive dimension. Let A, A* denote a tridiagonal pair on V with diameter d. We say that A, A* has Krawtchouk type whenever the sequence {d - 2 i}i = 0d is a standard ordering of the eigenvalues of A and a standard ordering of the eigenvalues of A*. Assume A, A* has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form 〈, 〉 on V such that 〈 Au, v 〉 = 〈 u, Av 〉 and 〈 A* u, v 〉 = 〈 u, A* v 〉 for u, v ∈ V. We show that the following tridiagonal pairs are isomorphic: (i) A, A*; (ii) - A, - A*; (iii) A*, A; (iv) - A*, - A. We give a number of related results and conjectures. © 2007 Elsevier Inc. All rights reserved.

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