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Some Finite Abelian Group Theory and Some \({q}\)-Series Identities

Annals of Combinatorics volume 20, pages 361–371 (2016)Cite this article

Abstract

The purpose of this article is to compute certain weighted sums over various subposets of the poset of isomorphism classes of finite abelian \({\ell}\)-groups, where \({\ell}\) is a fixed odd prime. These sums are similar to previous sums computed by Hall and Cohen-Lenstra. The computation expands upon the previous analysis of Cohen-Lenstra while also using the tools of hypergeometric functions and \({q}\)-series. The identities computed in this article, while aesthetically pleasing in their own right, also turn out to be applicable to the construction of a random matrix version of the Cohen-Lenstra-Martinet heuristics.

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Authors and Affiliations

  1. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, PO Box 751, Portland, OR, 97207-0751, USA

    Derek Garton

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  1. Derek Garton
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Correspondence to Derek Garton.

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Garton, D. Some Finite Abelian Group Theory and Some \({q}\)-Series Identities. Ann. Comb. 20, 361–371 (2016). https://doi.org/10.1007/s00026-016-0299-8

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  • DOI: https://doi.org/10.1007/s00026-016-0299-8

Mathematics Subject Classification

  • 05E15
  • 20K01

Keywords

  • posets
  • finite abelian groups
  • \({q}\)-series