2 7 M ay 2 00 7 Transition maps between the 24 bases for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → 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. 1 Leonard pairs We begin by recalling the notion of a Leonard pair. We will use the following terms. A square matrix X is said to be tridiagonal whenever each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. Assume X is tridiagonal. Then X is said to be irreducible whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. We now define a Leonard pair. For the rest of this paper K will denote a field. Definition 1.1 [41] Let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair A,A where A : V → V and A : V → V are linear transformations 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. Note 1.2 It is a common notational convention to use A to represent the conjugatetranspose of A. We are not using this convention. In a Leonard pair A,A the linear transformations A and A are arbitrary subject to (i) and (ii) above. We refer the reader to [9,11,25,28–34,36,39–41,43–50,52,54,55,57] for background on Leonard pairs. We especially recommend the survey [50]. See [1–8,10,12–24,26,27,35,37, 38,42,51,53,56] for related topics. 2 Leonard systems When working with a Leonard pair, it is convenient to consider a closely related object called a Leonard system. To prepare for our definition of a Leonard system, we recall a few concepts from linear algebra. Let d denote a nonnegative integer and let Matd+1(K) denote the K-algebra consisting of all d+ 1 by d + 1 matrices that have entries in K. We index the rows and columns by 0, 1, . . . , d. We let K denote the K-vector space of all d + 1 by 1 matrices that have entries in K. We index the rows by 0, 1, . . . , d. We view K d+1 as a left module for Matd+1(K). We observe this module is irreducible. For the rest of this paper, let A denote a K-algebra isomorphic to Matd+1(K) and let V denote an irreducible left A-module. We remark that V is unique up to isomorphism of A-modules, and that V has dimension d + 1. By a basis for V we mean a sequence of vectors that are linear independent and span V . We emphasize that the ordering is important. Let {vi} d i=0 denote a basis for V . For X ∈ A and Y ∈ Matd+1(K), we say Y represents X with respect to {vi} d i=0 whenever Xvj = ∑d i=0 Yijvi for 0 ≤ j ≤ d. For A ∈ A we say A is multiplicity-free whenever it has d + 1 mutually distinct eigenvalues in K. Assume A is multiplicity-free. Let {θi} d i=0 denote an ordering of the eigenvalues of A, and for 0 ≤ i ≤ d put Ei = ∏ 0≤j≤d j 6=i A− θjI θi − θj , (1) where I denotes the identity of A. We observe (i) AEi = θiEi (0 ≤ i ≤ d); (ii) EiEj = δi,jEi (0 ≤ i, j ≤ d); (iii) ∑d i=0 Ei = I; (iv) A = ∑d i=0 θiEi. Let D denote the subalgebra of A generated by A. Using (i)–(iv) we find the sequence {Ei} d i=0 is a basis for the K-vector space D. We call Ei the primitive idempotent of A associated with θi. It is helpful to think of these primitive idempotents as follows. Observe V = E0V + E1V + · · ·+ EdV (direct sum). For 0 ≤ i ≤ d, EiV is the (one dimensional) eigenspace of A in V associated with the eigenvalue θi, and Ei acts on V as the projection onto this eigenspace. By a Leonard pair in A we mean an ordered pair of elements taken from A that act on V as a Leonard pair in the sense of Definition 1.1. We now define a Leonard system.