The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

Han, Guo-Niu

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.

Nekrasov et Okounkov ont obtenu une nouvelle formule pour le développement des puissances du produit d’Euler, à l’aide des longueurs d’équerre des partitions d’entiers, dans leur étude de la théorie de Seiberg-Witten. Nous proposons un raffinement de cette formule reposant sur une propriété nouvelle des $t$-cores, qui permet de donner une démonstration élémentaire en faisant usage des identités de Macdonald. Nous obtenons aussi une extension, en ajoutant deux paramètres supplémentaires, qui peut être considérée comme une interpolation discrète entre les identités de Macdonald et la fonction génératrice des $t$-cores. Plusieurs applications en sont déduites, y compris la “formule d’équerre pointée”.

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of $t$-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for $t$-cores. Several applications are derived, including the “marked hook formula”.

Classification : 05A15, 05A17, 05A19, 11P82, 17B22
Keywords: Hook length, hook formula, partition, $t$-core, Euler product, Macdonald identities
Mot clés : longueur d’équerre, formule d’équerre, partition, $t$-core, produit d’Euler, identités de Macdonald

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