Double Lowering Operators on Polynomial

Abstract

Recently Sarah Bockting-Conrad introduced the double lowering operator $\psi$ for a tridiagonal pair. Motivated by $\psi$ we consider the following problem about polynomials. Let $\mathbb{F}$ denote an algebraically closed field. Let $x$ denote an indeterminate, and let $\mathbb{F}[x]$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb{F}$. Let $N$ denote a positive integer or $\infty$. Let $\{a_i{\}}_{i=0}^{N-1}$, $\{b_i{\}}_{i=0}^{N-1}$ denote scalars in $\mathbb{F}$ such that ${\sum }_{h=0}^{i-1}{a}_h\ne {\sum }_{h=0}^{i-1}{b}_h$ for $1\le i\le N$. For $0\le i\le N$ define polynomials ${\tau }_i,{\eta }_i\in \mathbb{F}[x]$ by ${\tau }_i={\prod }_{h=0}^{i-1}(x-a_h)$ and ${\eta }_i={\prod }_{h=0}^{i-1}(x-b_h)$. Let $V$ denote the subspace of $\mathbb{F}[x]$ spanned by $\{x^i{\}}_{i=0}^N$. An element $\psi \in \mathrm{End}(V)$ is called double lowering whenever $\psi {\tau }_i\in \mathbb{F}{\tau }_{i-1}$ and $\psi {\eta }_i\in \mathbb{F}{\eta }_{i-1}$ for $0\le i\le N$, where ${\tau }_{-1}=0$ and ${\eta }_{-1}=0$. We give necessary and sufficient conditions on $\{a_i{\}}_{i=0}^{N-1}$, $\{b_i{\}}_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.

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