Three mutually adjacent Leonard pairs

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 1There exist at most 3 mutually adjacent Leonard pairs on V provided the dimension of V is at least 2. Theorem 2Let (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 3Let (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.