Double Lowering Operators on Polynomial

Abstract

Recently Sarah Bockting-Conrad introduced the double lowering operator ψ for a tridiagonal pair. Motivated by ψ we consider the following problem about polynomials. Let F denote a field. Let x denote an indeterminate, and let F[x] denote the algebra consisting of the polynomials in x that have all coefficients in F. Let N denote a positive integer or ∞. Let. An element ψ ∈ End(V ) is called double lowering whenever ψτ i ∈ Fτ i−1 and ψη i ∈ Fη i−1 for 0 ≤ i ≤ N , where τ −1 = 0 and η −1 = 0. We give necessary and sufficient conditions on {a i } N −1 i=0 , {b i } N −1 i=0 for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.