Double Lowering Operators on Polynomials

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 an algebraically closed 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 {𝑎ᵢ}ᴺ⁻¹ᵢ₌₀, {𝑏ᵢ}ᴺ⁻¹ᵢ₌₀ denote scalars in 𝔽 such that ∑ⁱ⁻¹ₕ₌₀𝑎ₕ ≠ ∑ⁱ⁻¹ₕ₌₀𝑏ₕ for 1 ≤ 𝒾 ≤ 𝑁. For 0 ≤ 𝒾 ≤ 𝑁 define polynomials τᵢ, ηᵢ ∈ 𝔽[𝑥] by τᵢ=∏ⁱ⁻¹ₕ₌₀(𝑥−𝑎ₕ) and ηᵢ=∏ⁱ⁻¹ₕ₌₀(𝑥−𝑏ₕ). Let V denote the subspace of 𝔽[𝑥] spanned by {𝑥ᵢ}ᴺᵢ₌₀. An element ψ ∈ End(𝑉) is called double lowering whenever ψτᵢ ∈ 𝔽τᵢ₋₁ and ψηᵢ ∈ 𝔽ηᵢ₋₁ for 0 ≤ 𝒾 ≤ 𝑁, where τ₋₁ = 0 and η₋₁ = 0. We give necessary and sufficient conditions on {𝑎ᵢ}ᴺ⁻¹ᵢ₌₀, {𝑏ᵢ}ᴺ⁻¹ᵢ₌₀ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.