Three mutually adjacent Leonard pairs

Abstract

Let K denote a field of characteristic 0 and let V denote a vector space over K with positive finite dimension. Consider an ordered pair of linear transformations A : V → V and A ∗ : V → V that satisfies 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. Let ( A, A ∗) denote a Leonard pair on V. A basis for V is said to be standard for ( A, A ∗) whenever it satisfies (i) or (ii) above. A basis for V is said to be split for ( A, A ∗) whenever with respect to this basis the matrix representing one of A, A ∗ is lower bidiagonal and the matrix representing the other is upper bidiagonal. Let ( A, A ∗) and ( B, B ∗) denote Leonard pairs on V. We say these pairs are adjacent whenever each basis for V which is standard for ( A, A ∗) (resp. ( B, B ∗)) is split for ( B, B ∗) (resp. ( A, A ∗)). Our main results are as follows. Theorem 1 There exist at most 3 mutually adjacent Leonard pairs on V provided the dimension of V is at least 2. Theorem 2 Let (A, A ∗ ), (B, B ∗ ), and (C, C ∗ ) denote three mutually adjacent Leonard pairs on V. Then for each of these pairs, the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Theorem 3 Let (A, A ∗ ) denote a Leonard pair on V whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Then there exist Leonard pairs (B, B ∗ ) and (C, C ∗ ) on V such that (A, A ∗ ), (B, B ∗ ), and (C, C ∗ ) are mutually adjacent.