Double Lowering Operators on Polynomials
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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
| ISSN: | 1815-0659 |
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| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211179 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Double Lowering Operators on Polynomials. Paul Terwilliger. SIGMA 17 (2021), 009, 38 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | 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₌₀ₕ ≠ ∑ⁱ⁻¹ₕ₌₀ₕ 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.
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| ISSN: | 1815-0659 |