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Partition identities and quiver representations

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Abstract

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector.

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Acknowledgements

We thank Bruce Berndt and Ae Ja Yee for historical references concerning Durfee square identities. We also thank an anonymous referee for their helpful remarks. RR was supported in part by NSF Grant DMS-1200685. AW and AY were supported by a UIUC Campus Research Board and by an NSF grant.

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Authors and Affiliations

  1. Department of Mathematics, The University of North Carolina at Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, NC, 27599, USA

    Richárd Rimányi

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Anna Weigandt & Alexander Yong

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  1. Richárd Rimányi
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  2. Anna Weigandt
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  3. Alexander Yong
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Correspondence to Anna Weigandt.

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Rimányi, R., Weigandt, A. & Yong, A. Partition identities and quiver representations. J Algebr Comb 47, 129–169 (2018). https://doi.org/10.1007/s10801-017-0771-5

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  • DOI: https://doi.org/10.1007/s10801-017-0771-5

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