Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions
We present an explicit formula for the transition matrix from the type ₙ Koornwinder polynomials ₍₁ᵣ₎(|, , c, |, ) with one column diagrams, to the type ₙ monomial symmetric polynomials m₍₁ᵣ₎(). The entries of the matrix C enjoy a set of four-term recursion relations. These recursions provide us wi...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210764 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions. Ayumu Hoshino and Jun'ichi Shiraishi. SIGMA 16 (2020), 084, 28 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862739622297075712 |
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| author | Hoshino, Ayumu Shiraishi, Jun'ichi |
| author_facet | Hoshino, Ayumu Shiraishi, Jun'ichi |
| citation_txt | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions. Ayumu Hoshino and Jun'ichi Shiraishi. SIGMA 16 (2020), 084, 28 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present an explicit formula for the transition matrix from the type ₙ Koornwinder polynomials ₍₁ᵣ₎(|, , c, |, ) with one column diagrams, to the type ₙ monomial symmetric polynomials m₍₁ᵣ₎(). The entries of the matrix C enjoy a set of four-term recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from ₙ to ₙ₋₁. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: ₍₁ᵣ₎(|, , c, |, ) ⟷ ₍₁ᵣ₎(|, −, c, |, ) ⟷ ₍₁ᵣ₎(|, −, c, −c|, ) ⟷ ₍₁ᵣ₎(|¹/²c, −¹/²c, c, −c|, ) ⟷ ₍₁ᵣ₎(|¹/², −¹/², 1, −1|, ). We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type Bₙ, namely the transition matrix from the Schur polynomials ⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(|; , ) to the Hall-Littlewood polynomials ⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(|; 0, ). We also present a conjecture for the asymptotically free eigenfunctions of the ₙ -Toda operator, which can be regarded as a branching formula from the ₙ -Toda eigenfunction restricted to the ₙ₋₁ -Toda eigenfunctions.
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| first_indexed | 2026-04-17T17:28:55Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210764 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T17:28:55Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
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| spelling | Hoshino, Ayumu Shiraishi, Jun'ichi 2025-12-17T14:30:51Z 2020 Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions. Ayumu Hoshino and Jun'ichi Shiraishi. SIGMA 16 (2020), 084, 28 pages 1815-0659 2020 Mathematics Subject Classification: 33D52; 33D45 arXiv:2002.02148 https://nasplib.isofts.kiev.ua/handle/123456789/210764 https://doi.org/10.3842/SIGMA.2020.084 We present an explicit formula for the transition matrix from the type ₙ Koornwinder polynomials ₍₁ᵣ₎(|, , c, |, ) with one column diagrams, to the type ₙ monomial symmetric polynomials m₍₁ᵣ₎(). The entries of the matrix C enjoy a set of four-term recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from ₙ to ₙ₋₁. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: ₍₁ᵣ₎(|, , c, |, ) ⟷ ₍₁ᵣ₎(|, −, c, |, ) ⟷ ₍₁ᵣ₎(|, −, c, −c|, ) ⟷ ₍₁ᵣ₎(|¹/²c, −¹/²c, c, −c|, ) ⟷ ₍₁ᵣ₎(|¹/², −¹/², 1, −1|, ). We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type Bₙ, namely the transition matrix from the Schur polynomials ⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(|; , ) to the Hall-Littlewood polynomials ⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(|; 0, ). We also present a conjecture for the asymptotically free eigenfunctions of the ₙ -Toda operator, which can be regarded as a branching formula from the ₙ -Toda eigenfunction restricted to the ₙ₋₁ -Toda eigenfunctions. Research of A.H. is supported by JSPS KAKENHI (Grant Numbers 16K05186 and 19K03530). Research of J.S. is supported by JSPS KAKENHI (Grant Numbers 15K04808, 19K03512, 16K05186, and 19K03530). The authors thank M. Noumi and L. Rybnikov for stimulating discussion. They also thank the anonymous referees for valuable comments and suggestions, including the proof of Theorem 3.3 based on Bressoud's matrix inversion. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions Article published earlier |
| spellingShingle | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions Hoshino, Ayumu Shiraishi, Jun'ichi |
| title | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions |
| title_full | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions |
| title_fullStr | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions |
| title_full_unstemmed | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions |
| title_short | Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions |
| title_sort | branching rules for koornwinder polynomials with one column diagrams and matrix inversions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210764 |
| work_keys_str_mv | AT hoshinoayumu branchingrulesforkoornwinderpolynomialswithonecolumndiagramsandmatrixinversions AT shiraishijunichi branchingrulesforkoornwinderpolynomialswithonecolumndiagramsandmatrixinversions |