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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References
Andrews, G.E., Eriksson, K.: Integer Partitions. Cambridge University Press, Cambridge (2004)
Apostol, T.: Modular Functions and Dirichlet Series in Number Theory, 2nd edn. Springer, Berlin (1990)
Atiyah, M.: The logarithm of the Dedekind \(\eta \)-function. Math. Ann. 278, 335–380 (1987)
Comter, L.: Advanced Combinatorics, Enlarged edn. D. Reidel Publishing Co., Dordrecht (1974)
D’Arcais, F.: Développement en série. Intermédiaire Math. 20, 233–234 (1913)
Darmon, H.: Andrew Wiles’ marvellous proof. EMS Newsl. 104, 7–13 (2017)
Fulton, W.: Young Tableaux. Cambridge University Press, Cambridge (1997)
Garvan, F., Kim, D., Stanton, D.: Cranks and t-cores. Invent. Math. 101, 1–17 (1990)
Han, G.: The Nekrasov–Okounkov hook length formula: refinement, elementary proof and applications. Ann. Inst. Fourier (Grenoble) 60(1), 1–29 (2010)
Heim, B., Luca, F., Neuhauser, M.: On cyclotomic factors of polynomials related to modular forms. Ramanujan J. 48, 445–458 (2019)
Heim, B., Neuhauser, M., Weisse, A.: Records on the vanishing of Fourier coefficients of powers of the Dedekind eta function. Res. Number Theory 4, 32 (2018)
Heim, B., Neuhauser, M.: On conjectures regarding the Nekrasov–Okounkov hook length formula. Archiv der Mathematik 113, 355–366 (2019)
Heim, B., Neuhauser, M.: Log-concavity of recursively defined polynomials. J. Integer Seq. 22, 1–12 (2019)
Köhler, G.: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Berlin (2011)
Kostant, B.: On Macdonald’s \(\eta \)-function formula, the Laplacian and generalized exponents. Adv. Math. 20, 179–212 (1976)
Kostant, B.: Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra. Invent. Math. 158, 181–226 (2004)
Lehmer, D.: The vanishing of Ramanujan’s \(\tau \left( n\right) \). Duke Math. J. 14, 429–433 (1947)
Macdonald, I.G.: Affine root systems and Dedekind’s \(\eta \)-function. Invent. Math. 15, 91–143 (1972)
Murty, M., Murty, V.: The Mathematical Legacy of Srinivasa Ramanujan. Springer, New Delhi (2013)
Nekrasov, N., Okounkov, A.: Seiberg–Witten Theory and Random Partitions. The Unity of Mathematics. Progress in Mathematics, vol. 244. Birkhäuser, Boston (2006)
Newman, M.: An identity for the coefficients of certain modular forms. J. Lond. Math. Soc. 30, 488–493 (1955)
Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series. Conference Board of Mathematical Sciences, vol. 102. American Mathematical Society, Providence (2003)
Serre, J.: Sur la lacunarité des puissances de \(\eta \). Glasgow Math. J. 27, 203–221 (1985)
Westbury, B.: Universal characters from the Macdonald identities. Adv. Math. 202(1), 50–63 (2006)
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We thank both reviewers for carefully reading the manuscript and useful comments.
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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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DOI: https://doi.org/10.1007/s40687-019-0201-5