2 8 Ju n 20 04 Leonard pairs from 24 points of view ∗

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal. (ii) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal. We call such a pair a Leonard pair on V . Referring to the above Leonard pair, we investigate 24 bases for V on which the action of A and A takes an attractive form. Our bases are described as follows. Let Ω denote the set consisting of four symbols 0, d, 0, d. We identify the symmetric group S4 with the set of all linear orderings of Ω. For each element g of S4, we define an (ordered) basis for V , which we denote by [g]. The 24 resulting bases are related as follows. For all elements wxyz in S4, the transition matrix from the basis [wxyz] to the basis [xwyz] (resp. [wyxz]) is diagonal (resp. lower triangular). The basis [wxzy] is the basis [wxyz] in inverted order. The transformations A and A act on the 24 bases as follows. For all g ∈ S4, let A g (resp. A∗g) denote the matrix representing A (resp. A) with respect to [g]. To describe Ag and A∗g, we refer to 0, d as the starred elements of Ω. Writing g = wxyz, if neither of y, z are starred then Ag is diagonal and A∗g is irreducible tridiagonal. If y is starred but z not, then Ag is lower bidiagonal and A∗g is upper bidiagonal. If z is starred but y not, then Ag is upper bidiagonal and A∗g is lower bidiagonal. If both of y, z are starred, then Ag is irreducible tridiagonal and A∗g is diagonal. We define a symmetric binary relation on S4 called adjacency. An element wxyz of S4 is by definition adjacent to each of xwyz, wyxz, wxzy and no other elements of S4. For all ordered pairs of adjacent elements g, h in S4, we find the entries of the transition matrix from the basis [g] to the basis [h]. We express these entries in terms of the eigenvalues of A, the eigenvalues of A, and two sequences of parameters called the first split sequence and the second split sequence. For all g ∈ S4, we compute the entries of Ag and A∗g in terms of the eigenvalues of A, the eigenvalues of A, the first split sequence and the second split sequence. ∗