Three mutually adjacent Leonard pairs

Let denote a field of characteristic 0 and let V denote a vector space over 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.