Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas
Abstract
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 infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive -Nekrasov--Okounkov formulas associated to each type. The latter for limit cases of yield Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald.
Keywords
Cite
@article{arxiv.2306.08071,
title = {Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas},
author = {David Wahiche},
journal= {arXiv preprint arXiv:2306.08071},
year = {2026}
}
Comments
Major rework. Details have been added. A lot of typos have been now fixed and some of the results in type D were incorrect so it has now been fixed. The organization of the paper has been changed based on recommendations. An interested reader can find a Sagemath version coded by the author to check some of the results of Section 4 and 6