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Mathematics > Combinatorics

arXiv:1808.01567 (math)
[Submitted on 5 Aug 2018 (v1), last revised 28 Nov 2018 (this version, v2)]

Title:Combinatorial cluster expansion formulas from triangulated surfaces

Authors:Toshiya Yurikusa
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Abstract:We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between perfect matchings of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.
Comments: 31 pages
Subjects: Combinatorics (math.CO); Commutative Algebra (math.AC); Representation Theory (math.RT)
MSC classes: 13F60, 05C70, 05E15
Cite as: arXiv:1808.01567 [math.CO]
  (or arXiv:1808.01567v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1808.01567
Journal reference: Electron. J. Combin. Vol. 26, 2 (2019) P2.33
Related DOI: https://doi.org/10.37236/8351

Submission history

From: Toshiya Yurikusa [view email]
[v1] Sun, 5 Aug 2018 06:46:50 UTC (43 KB)
[v2] Wed, 28 Nov 2018 17:07:49 UTC (44 KB)
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