Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials
The Cauchy identities play an important role in the theory of symmetric functions. It is known that Cauchy sums for the -Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a -Pochhammer symbol. We consider sums with restrictions on the length of the first rows of...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212356 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials. Takashi Imamura, Matteo Mucciconi and Tomohiro Sasamoto. SIGMA 20 (2024), 064, 28 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862712858344685568 |
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| author | Imamura, Takashi Mucciconi, Matteo Sasamoto, Tomohiro |
| author_facet | Imamura, Takashi Mucciconi, Matteo Sasamoto, Tomohiro |
| citation_txt | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials. Takashi Imamura, Matteo Mucciconi and Tomohiro Sasamoto. SIGMA 20 (2024), 064, 28 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Cauchy identities play an important role in the theory of symmetric functions. It is known that Cauchy sums for the -Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a -Pochhammer symbol. We consider sums with restrictions on the length of the first rows of the labels for both polynomials and prove an identity relating them. The proof is based on techniques from integrable probability: we rewrite the identity in terms of two probability measures: the -Whittaker measure and the periodic Schur measure. The relation follows by comparing their Fredholm determinant formulas.
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| first_indexed | 2026-03-19T19:49:03Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212356 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T19:49:03Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Imamura, Takashi Mucciconi, Matteo Sasamoto, Tomohiro 2026-02-05T09:56:25Z 2024 Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials. Takashi Imamura, Matteo Mucciconi and Tomohiro Sasamoto. SIGMA 20 (2024), 064, 28 pages 1815-0659 2020 Mathematics Subject Classification: 05A19; 05E05; 60J10 arXiv:2106.11913 https://nasplib.isofts.kiev.ua/handle/123456789/212356 https://doi.org/10.3842/SIGMA.2024.064 The Cauchy identities play an important role in the theory of symmetric functions. It is known that Cauchy sums for the -Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a -Pochhammer symbol. We consider sums with restrictions on the length of the first rows of the labels for both polynomials and prove an identity relating them. The proof is based on techniques from integrable probability: we rewrite the identity in terms of two probability measures: the -Whittaker measure and the periodic Schur measure. The relation follows by comparing their Fredholm determinant formulas. This work was initiated during the MATRIX program ”Non-equilibrium systems and special functions” in 2018. The authors are grateful to Dan Betea and Jérémie Bouttier for the introduction to their recent work [6] and for discussions about the periodic Schur measure and the free Fermion at positive temperature during the program. In particular, TS thanks DB for pointing out I.5,28(a) of Macdonald’s book. We greatly appreciate the creative atmosphere and warm hospitality of the MATRIX institute. We are grateful to Alexei Borodin and Guillaume Barraquand, whose conversations led to Section 4 and to a better understanding of relations between our result and those of [9]. We thank the anonymous referees for careful reading of the manuscript and helpful comments. The work of TI has been supported by JSPS KAKENHI Grant No. JP16K05192, No. JP19H01793, No. JP20K03626, and No. JP22H01143. The work of TS has been supported by JSPS KAKENHI Grants No. JP15K05203, No. JP16H06338, No. JP18H01141, No. JP18H03672, No. JP19L03665, No. JP21H04432, No. JP22H01143. The work of MM has been partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101030938. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials Article published earlier |
| spellingShingle | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials Imamura, Takashi Mucciconi, Matteo Sasamoto, Tomohiro |
| title | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials |
| title_full | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials |
| title_fullStr | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials |
| title_full_unstemmed | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials |
| title_short | Identity between Restricted Cauchy Sums for the -Whittaker and Skew Schur Polynomials |
| title_sort | identity between restricted cauchy sums for the -whittaker and skew schur polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212356 |
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