un 2 00 7 Tridiagonal pairs of Krawtchouk type

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 − 2i}i=0 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 〈Au, v〉 = 〈u,Av〉 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.