Double Lowering Operators on Polynomials

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2021
• Автор: Terwilliger, Paul
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2021
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/211179
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Double Lowering Operators on Polynomials. Paul Terwilliger. SIGMA 17 (2021), 009, 38 pages

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 denote an algebraically closed field. Let denote an indeterminate, and let [ ] denote the algebra consisting of the polynomials in that have all coefficients in . Let denote a positive integer or ∞. Let { ᵢ}ᴺ⁻¹ᵢ₌₀, { ᵢ}ᴺ⁻¹ᵢ₌₀ denote scalars in such that ∑ⁱ⁻¹ₕ₌₀ ₕ ≠ ∑ⁱ⁻¹ₕ₌₀ ₕ for 1 ≤ ≤ . For 0 ≤ ≤ define polynomials τᵢ, ηᵢ ∈ [ ] by τᵢ=∏ⁱ⁻¹ₕ₌₀( − ₕ) and ηᵢ=∏ⁱ⁻¹ₕ₌₀( − ₕ). Let V denote the subspace of [ ] spanned by { ᵢ}ᴺᵢ₌₀. An element ψ ∈ End( ) is called double lowering whenever ψτᵢ ∈ τᵢ₋₁ and ψηᵢ ∈ ηᵢ₋₁ for 0 ≤ ≤ , where τ₋₁ = 0 and η₋₁ = 0. We give necessary and sufficient conditions on { ᵢ}ᴺ⁻¹ᵢ₌₀, { ᵢ}ᴺ⁻¹ᵢ₌₀ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.