  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 $A,A^*$.

Rings and Algebras;Combinatorics