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

Guo-Niu Han[1]

  • [1] IRMA, UMR 7501 Université de Strasbourg et CNRS 7 rue René-Descartes 67084 Strasbourg (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 1, page 1-29
  • ISSN: 0373-0956

Abstract

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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”.

How to cite

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Han, Guo-Niu. "The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications." Annales de l’institut Fourier 60.1 (2010): 1-29. <http://eudml.org/doc/116266>.

@article{Han2010,
abstract = {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”.},
affiliation = {IRMA, UMR 7501 Université de Strasbourg et CNRS 7 rue René-Descartes 67084 Strasbourg (France)},
author = {Han, Guo-Niu},
journal = {Annales de l’institut Fourier},
keywords = {Hook length; hook formula; partition; $t$-core; Euler product; Macdonald identities; hook length; -core; MacDonald identities},
language = {eng},
number = {1},
pages = {1-29},
publisher = {Association des Annales de l’institut Fourier},
title = {The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications},
url = {http://eudml.org/doc/116266},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Han, Guo-Niu
TI - The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 1
SP - 1
EP - 29
AB - 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”.
LA - eng
KW - Hook length; hook formula; partition; $t$-core; Euler product; Macdonald identities; hook length; -core; MacDonald identities
UR - http://eudml.org/doc/116266
ER -

References

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