Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple

Abstract

Recently, Paul Terwilliger introduced the notion of a lowering–raising (or LR) triple, and classified the LR triples. An LR triple is defined as follows. Fix an integer d≥0, a field F, and a vector space V over F with dimension d+1. By a decomposition of V we mean a sequence {Vi}i=0d of 1-dimensional subspaces of V whose sum is V. For a linear transformation A from V to V, we say A lowers {Vi}i=0d whenever AVi=Vi−1 for 0≤i≤d, where V−1=0. We say A raises {Vi}i=0d whenever AVi=Vi+1 for 0≤i≤d, where Vd+1=0. An ordered pair of linear transformations A, B from V to V is called LR whenever there exists a decomposition {Vi}i=0d of V that is lowered by A and raised by B. In this case the decomposition {Vi}i=0d is uniquely determined by A, B; we call it the (A,B)-decomposition of V. Consider a 3-tuple of linear transformations A, B, C from V to V such that any two of A, B, C form an LR pair on V. Such a 3-tuple is called an LR triple on V. Let α, β, γ be nonzero scalars in F. The triple αA, βB, γC is an LR triple on V, said to be associated to A, B, C. Let {Vi}i=0d be a decomposition of V and let X be a linear transformation from V to V. We say X is tridiagonal with respect to {Vi}i=0d whenever XVi⊆Vi−1+Vi+Vi+1 for 0≤i≤d. Let X be the vector space over F consisting of the linear transformations from V to V that are tridiagonal with respect to the (A,B) and (B,C) and (C,A) decompositions of V. There is a special class of LR triples, called q-Weyl type. In the present paper, we find a basis of X for each LR triple that is not associated to an LR triple of q-Weyl type.