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 |
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| Format: | Artikel |
| Sprache: | Englisch |
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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 |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862670247762329600 |
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| author | Terwilliger, Paul |
| author_facet | Terwilliger, Paul |
| citation_txt | Double Lowering Operators on Polynomials. Paul Terwilliger. SIGMA 17 (2021), 009, 38 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2026-03-16T14:12:23Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211179 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T14:12:23Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Terwilliger, Paul 2025-12-25T13:23:40Z 2021 Double Lowering Operators on Polynomials. Paul Terwilliger. SIGMA 17 (2021), 009, 38 pages 1815-0659 2020 Mathematics Subject Classification: 33D15; 15A21 arXiv:2003.09666 https://nasplib.isofts.kiev.ua/handle/123456789/211179 https://doi.org/10.3842/SIGMA.2021.009 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. The author would like to thank Kazumasa Nomura for giving this paper a close reading and offering many valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Double Lowering Operators on Polynomials Article published earlier |
| spellingShingle | Double Lowering Operators on Polynomials Terwilliger, Paul |
| title | Double Lowering Operators on Polynomials |
| title_full | Double Lowering Operators on Polynomials |
| title_fullStr | Double Lowering Operators on Polynomials |
| title_full_unstemmed | Double Lowering Operators on Polynomials |
| title_short | Double Lowering Operators on Polynomials |
| title_sort | double lowering operators on polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211179 |
| work_keys_str_mv | AT terwilligerpaul doubleloweringoperatorsonpolynomials |