David Wahiche: Macdonald identities, affine Grassmannian elements and hook length formulae

Abstract: The Nekrasov--Okounkov formula provides an expression of Fourier coefficients of powers of the Euler function as a sum of product of hook lengths. The aim of this talk is to show how formulas of these kind can be derived from a specialization of the Macdonald identity, also called Weyl--Kac denominator formula. The latter can be rewritten as a sum indexed by affine Grassmannian elements, where appears the atomic length introduced by Chapelier-Laget and Gerber. I will sketch the dictionary in type A between these elements and some subsets of integer partitions. This is an introduction to a joint work with Cédric Lecouvey (https://arxiv.org/abs/2404.10532).