Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers
We present an explicit formula for the transition matrix C from the type Cₙ degeneration of the Koornwinder polynomials P₍₁ᵣ₎(x|a,−a,c,−c|q,t) with one column diagrams, to the type Cₙ monomial symmetric polynomials m₍₁ᵣ₎(x). The entries of the matrix C enjoy a set of three-term recursion relations,...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209856 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers / A. Hoshino, J. Shiraishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 22 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862645556113833984 |
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| author | Hoshino, A. Shiraishi, J. |
| author_facet | Hoshino, A. Shiraishi, J. |
| citation_txt | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers / A. Hoshino, J. Shiraishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present an explicit formula for the transition matrix C from the type Cₙ degeneration of the Koornwinder polynomials P₍₁ᵣ₎(x|a,−a,c,−c|q,t) with one column diagrams, to the type Cₙ monomial symmetric polynomials m₍₁ᵣ₎(x). The entries of the matrix C enjoy a set of three-term recursion relations, which can be regarded as a (a, c, t)-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type (Cₙ, Cₙ) Macdonald polynomials P⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|b;q,t)=P₍₁ᵣ₎(x|b¹ᐟ²,−b¹ᐟ²,q¹ᐟ²b¹ᐟ²,−q¹ᐟ²b¹ᐟ²|q,t). It is also shown that the q-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials P⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|q;q,q) to the Hall-Littlewood polynomials PP⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|t;0,t).
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| first_indexed | 2025-12-07T14:26:49Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209856 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T14:26:49Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
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| spelling | Hoshino, A. Shiraishi, J. 2025-11-27T14:56:40Z 2018 Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers / A. Hoshino, J. Shiraishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D52; 33D45 arXiv: 1801.09939 https://nasplib.isofts.kiev.ua/handle/123456789/209856 https://doi.org/10.3842/SIGMA.2018.101 We present an explicit formula for the transition matrix C from the type Cₙ degeneration of the Koornwinder polynomials P₍₁ᵣ₎(x|a,−a,c,−c|q,t) with one column diagrams, to the type Cₙ monomial symmetric polynomials m₍₁ᵣ₎(x). The entries of the matrix C enjoy a set of three-term recursion relations, which can be regarded as a (a, c, t)-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type (Cₙ, Cₙ) Macdonald polynomials P⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|b;q,t)=P₍₁ᵣ₎(x|b¹ᐟ²,−b¹ᐟ²,q¹ᐟ²b¹ᐟ²,−q¹ᐟ²b¹ᐟ²|q,t). It is also shown that the q-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials P⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|q;q,q) to the Hall-Littlewood polynomials PP⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|t;0,t). Research of A.H. is supported by JSPS KAKENHI (Grant Number 16K05186). Research of J.S. is supported by JSPS KAKENHI (Grant Numbers 15K04808 and 16K05186). The authors thank M. Noumi, B. Feigin, and H. Awata for stimulating discussion. They thank the anonymous referees for their various constructive comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers Article published earlier |
| spellingShingle | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers Hoshino, A. Shiraishi, J. |
| title | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers |
| title_full | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers |
| title_fullStr | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers |
| title_full_unstemmed | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers |
| title_short | Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers |
| title_sort | macdonald polynomials of type cₙ with one-column diagrams and deformed catalan numbers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209856 |
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