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The Dedekind eta function and D’Arcais-type polynomials

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Abstract

D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend results of Kostant and Han.

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Acknowledgements

We thank both reviewers for carefully reading the manuscript and useful comments.

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Correspondence to Bernhard Heim.

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Heim, B., Neuhauser, M. The Dedekind eta function and D’Arcais-type polynomials. Res Math Sci 7, 3 (2020). https://doi.org/10.1007/s40687-019-0201-5

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Keywords

  • Dedekind eta function
  • Nekrasov–Okounkov
  • Partitions
  • Polynomials
  • Recursions

Mathematics Subject Classification

  • Primary 11F20
  • 26C10
  • Secondary 05A17
  • 11F11