A ] 1 9 Fe b 20 06 Matrix units associated with the split basis of a Leonard pair Kazumasa Nomura and

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 (i), (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 . It is known that there exists a basis for V with respect to which the matrix representing A is lower bidiagonal and the matrix representing A∗ is upper bidiagonal. In this paper we give some formulae involving the matrix units associated with this basis. 1 Leonard pairs and Leonard systems 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 [18] 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), (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 [3], [9], [12], [13], [14], [15], [16], [17], [18], [20], [21], [22], [23], [24], [25], [26], [27], [29], [30] for background on Leonard pairs. We especially recommend the survey [27]. See [1], [2], [4], [5], [6], [7], [8], [10], [11], [19], [28] for related topics. It is known that there exists a basis for V with respect to which the matrix representing A is lower bidiagonal and the matrix representing A is upper bidiagonal [18, Theorem 3.2]. In this paper we give some formulae involving the matrix units associated with this basis. We also display some related formulae involving the primitive idempotents of A and A, which might be of independent interest. 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. For the rest of this paper we let A denote a K-algebra isomorphic to Matd+1(K). Let V denote a simple A-module. We remark that V is unique up to isomorphism of A-modules, and that V has dimension d + 1. Let v0, v1, . . . , vd denote a basis for V . For X ∈ A and Y ∈ Matd+1(K), we say Y represents X with respect to v0, v1, . . . , vd 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 θ0, θ1, . . . , θd 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. We call Ei the primitive idempotent of A associated with θi. By a Leonard pair in A we mean an ordered pair of elements in A that act on V as a Leonard pair in the sense of Definition 1.1. We now define a Leonard system. Definition 1.3 [18] By a Leonard system in A we mean a sequence (A; {Ei} d i=0;A ; {E i } d i=0) that satisfies (i)–(v) below. (i) Each of A, A is a multiplicity-free element in A. (ii) E0, E1, . . . , Ed is an ordering of the primitive idempotents of A. (iii) E 0 , E ∗ 1 , . . . , E ∗ d is an ordering of the primitive idempotents of A . (iv) For 0 ≤ i, j ≤ d, EiA Ej = { 0 if |i− j| > 1, 6= 0 if |i− j| = 1.