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Dungeons and Dragons: Combinatorics for the \(\varvec{dP_3}\) Quiver

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Abstract

In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work (Zhang in Cluster variables and perfect matchings of subgraphs of the \(dP_{3}\) lattice, http://www.math.umn.edu/~reiner/REU/Zhang2012.pdf. arXiv:1511.0655, 2012; Leoni et al. in J Phys A Math Theor 47:474011, 2014), which arose during the second author’s mentoriship of undergraduates, and more recently of both authors (Lai and Musiker in Commun Math Phys 356(3):823–881, 2017), analyzed the cluster algebra associated with the cone over \(\mathbf {dP_3}\), the del Pezzo surface of degree 6 (\({\mathbb {C}}{\mathbb {P}}^2\) blown up at three points). By investigating sequences of toric mutations, those occurring only at vertices with two incoming and two outgoing arrows, in this cluster algebra, we obtained a family of cluster variables that could be parameterized by \({\mathbb {Z}}^3\) and whose Laurent expansions had elegant combinatorial interpretations in terms of dimer partition functions (in most cases). While the earlier work (Lai and Musiker 2017; Zhang 2012; Leoni et al. 2014) focused exclusively on one possible initial seed for this cluster algebra, there are in total four relevant initial seeds (up to graph isomorphism). In the current work, we explore the combinatorics of the Laurent expansions from these other initial seeds and how this allows us to relate enumerations of perfect matchings on Dungeons to Dragons.

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Notes

  1. All four toric phases of the \(\mathbf {dP_3}\) quiver appeared for the first time in [10, Section 6.3], but this required brute force, since it preceded the more efficient dimer model technology.

  2. Our construction is defined for those cases where the contour defined by lifting the point in \({\mathbb {Z}}^3\) to \({\mathbb {Z}}^6\) has no self-intersections.

  3. Research Experiences for Undergraduates (REU) is an NSF-funded program. See more details about this NSF program in the link https://www.nsf.gov/crssprgm/reu/, and about the REU program in combinatorics at University of Minnesota in the link http://www-users.math.umn.edu/~reiner/REU/REU.html.

  4. This description holds for five out of the six graph families, except for \(F^{(3)}\) (as we detail below). This exception is inherited from the description from [30].

  5. This is because of its cluster algebraic interpretation or by applying the Laurent Phenomenon [13].

  6. Sometimes, this is called the “rainbow condition” and agrees with the tripartite pairing boundary condition of [23, Sec. 6]; we thank Helen Jenne for turning our attention to [23].

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Acknowledgements

We are grateful to Richard Eager, Sebastian Franco, Rick Kenyon, and David Speyer for numerous helpful discussions. We thank the referees for their helpful comments. We also used the cluster algebras package [38] in Sage [41] for numerous computations.

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  1. Department of Mathematics, University of Nebraska, Lincoln, NE, 68588, USA

    Tri Lai

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Gregg Musiker

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Correspondence to Gregg Musiker.

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Tri Lai was supported by the Simons Foundation Collaboration Grant-585923. Gregg Musiker was supported by NSF Grants DMS-13692980 and DMS-1148634.

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Lai, T., Musiker, G. Dungeons and Dragons: Combinatorics for the \(\varvec{dP_3}\) Quiver. Ann. Comb. 24, 257–309 (2020). https://doi.org/10.1007/s00026-019-00487-y

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