Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements
The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type Weyl denominator formula and the use of a polynomial argument. In this pa...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/213190 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements. Cédric Lecouvey and David Wahiche. SIGMA 21 (2025), 023, 45 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862599249554833408 |
|---|---|
| author | Lecouvey, Cédric Wahiche, David |
| author_facet | Lecouvey, Cédric Wahiche, David |
| citation_txt | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements. Cédric Lecouvey and David Wahiche. SIGMA 21 (2025), 023, 45 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of -core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas.
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| first_indexed | 2026-03-21T11:59:30Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-213190 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:59:30Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Lecouvey, Cédric Wahiche, David 2026-02-16T16:36:02Z 2025 Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements. Cédric Lecouvey and David Wahiche. SIGMA 21 (2025), 023, 45 pages 1815-0659 2020 Mathematics Subject Classification: 05E05; 05E10; 11P81 arXiv:2404.10532 https://nasplib.isofts.kiev.ua/handle/123456789/213190 https://doi.org/10.3842/SIGMA.2025.023 The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of -core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas. The authors would like to thank the referees as well as the editors for their precious comments and suggestions to help this paper to be much clearer. The authors would also like to thank Olivier Brunat, Nathan Chapelier-Laget, Thomas Gerber, Igor Haladjian, Guo-Niu Han, and Ole Warnaar for their interest and their precious remarks, as well as Philippe Nadeau for suggesting the reformulation of Theorem 7.4. Both authors are supported by the Agence Nationale de la Recherche funding ANR CORTIPOM 21-CE40-001. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements Article published earlier |
| spellingShingle | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements Lecouvey, Cédric Wahiche, David |
| title | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements |
| title_full | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements |
| title_fullStr | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements |
| title_full_unstemmed | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements |
| title_short | Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements |
| title_sort | macdonald identities, weyl-kac denominator formulas and affine grassmannian elements |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213190 |
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