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 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.