Compatibility and companions for Leonard pairs

Abstract

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let F denote a field, and let V denote a vector space over F with finite positive dimension. A Leonard pair on V is an ordered pair of diagonalizable F-linear maps A : V → V and A * : V → V that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs A, A * and B, B * on V are said to be compatible wheneverFor a Leonard pair A, A * on V , by a companion of A, A * we mean an F-linear map K : V → V such that K is a polynomial in A * and A − K, A * is a Leonard pair on V . The concepts of compatibility and companion are related as follows. For compatible Leonard pairs A, A * and B, B * on V , define K = A − B. Then K is a companion of A, A * . For a Leonard pair A, A * on V and a companion K of A, A * , define B = A − K and B * = A * . Then B, B * is a Leonard pair on V that is compatible with A, A * . Let A, A * denote a Leonard pair on V . We find all the Leonard pairs B, B * on V that are compatible with A, A * . For each solution B, B * we describe the corresponding companion K = A − B.