Tridiagonal pairs of Krawtchouk type Original Research Article

Let image denote an algebraically closed field with characteristic 0 and let V denote a vector space over image 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 image 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 image on V such that left angle bracketAu,vright-pointing angle bracket=left angle bracketu,Avright-pointing angle bracket and left angle bracketA*u,vright-pointing angle bracket=left angle bracketu,A*vright-pointing angle bracket for u,vset membership, variantV. 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.