Balanced Leonard Pairs

Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy the following two conditions: (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. Let v∗ 0 , v∗ 1 , . . . , v∗ d (respectively v0, v1, . . . , vd) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ≤ i ≤ d, let ai denote the coefficient of v ∗ i , when we write Av∗ i as a linear combination of v∗ 0 , v∗ 1 , . . . , v∗ d , and let a∗ i denote the coefficient of vi, when we write A ∗vi as a linear combination of v0, v1, . . . , vd. In this paper we show a0 = ad if and only if a ∗ 0 = a∗ d . Moreover we show that for d ≥ 1 the following are equivalent; (i) a0 = ad and a1 = ad−1; (ii) a ∗ 0 = a∗ d and a∗ 1 = a∗ d−1 ; (iii) ai = ad−i and a∗ i = a∗ d−i for 0 ≤ i ≤ d. These give a proof of a conjecture by the second author. We say A, A∗ is balanced whenever ai = ad−i and a ∗ i = a∗ d−i for 0 ≤ i ≤ d. We say A, A∗ is essentially bipartite (respectively essentially dual bipartite) whenever ai (respectively a ∗ i ) is independent of i for 0 ≤ i ≤ d. Observe that if A, A∗ is essentially bipartite or dual bipartite, then A, A∗ is balanced. For d 6= 2, we show that if A, A∗ is balanced then A, A∗ is essentially bipartite or dual bipartite.