Transition maps between the 24 bases for a Leonard pair

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In an earlier paper we described 24 special bases for $V$. One feature of these bases is that with respect to each of them the matrices that represent $A$ and $A^*$ are (i) diagonal and irreducible tridiagonal or (ii) irreducible tridiagonal and diagonal or (iii) lower bidiagonal and upper bidiagonal or (iv) upper bidiagonal and lower bidiagonal. For each ordered pair of bases among the 24, there exists a unique linear transformation from $V$ to $V$ that sends the first basis to the second basis; we call this the transition map. In this paper we find each transition map explicitly as a polynomial in $A,A^*$.