A crank-based approach to the theory of $3$-core partitions
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- by Olivier Brunat and Rishi Nath PDF
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Abstract:
This note is concerned with the set of integral solutions of the equation $x^2+3y^2=12n+4$, where $n$ is a positive integer. We will describe a parametrization of this set using the $3$-core partitions of $n$. In particular we construct a crank using the action of a suitable subgroup of the isometric group of the plane that we connect with the unit group of the ring of Eisenstein integers. We also show that the process goes in the reverse direction: from the solutions of the equation and the crank, we can describe the $3$-core partitions of $n$. As a consequence we describe an explicit bijection between $3$-core partitions and ideals of the ring of Eisenstein integers, explaining a result of G. Han and K. Ono obtained using modular forms.References
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Additional Information
- Olivier Brunat
- Affiliation: Université de Paris 7, Institut de mathématiques de Jussieu – Paris Rive Gauche, UFR de mathématiques, Case 7012, 75205 Paris Cedex 13, France
- MR Author ID: 740465
- ORCID: 0000-0003-4963-404X
- Email: olivier.brunat@imj-prg.fr
- Rishi Nath
- Affiliation: Department of Mathematics and Computer Science, York College, City University of New York, 94–20 Guy R. Brewer Blvd, Jamaica, New York 11435
- MR Author ID: 845320
- Email: rnath@york.cuny.edu
- Received by editor(s): November 15, 2020
- Received by editor(s) in revised form: December 4, 2020, and December 14, 2020
- Published electronically: October 25, 2021
- Communicated by: Romyar T. Sharifi
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 15-29
- MSC (2020): Primary 05A17, 11P83
- DOI: https://doi.org/10.1090/proc/15748
- MathSciNet review: 4335853