Pré-Publication, Document De Travail Année : 2023

Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas

Résumé

We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This yields on the other hand a combinatorial interpretation of the Macdonald identities for affine root systems of the $7$ infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov--Okounkov formulas associated to each type. The latter for limit cases of $q$ yield Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald.

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hal-04131809 , version 1 (15-05-2026)

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David Wahiche. Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas. 2026. ⟨hal-04131809⟩
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