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Higher Dimers on Snake Graphs and Continued Fractions

Algebra & Discrete Mathematics

Speaker: Sylvester Zhang, University of Minnesota
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Location: 1147 MSB
Start time: Mon, Jun 5 2023, 4:10PM

Snake graphs are certain planer bipartite graph that play important roles in the theory of cluster algebras. In particular, perfect matchings (a.k.a dimer covers) of snake graphs are in bijection with certain cluster variables. On the other side, work of Schiffler and Canakci showed that a continued fraction can be interpreted as a ratio of numbers of matchings of snake graphs. We investigate a higher dimensional analogue by counting ``higher-dimers’’ of snake graphs. In particular, we recover the definition of higher dimensional Fibonacci numbers of Raney, which converge to diagonal lengths of a regular polygon. We show that the number of m-dimers of a snake graph can be computed using a product of SL_{m+1} matrices, which yields as a by product a new notion of higher-dimensional continued fraction. If time permits, I will also discuss connections to other combinatorial objects like reverse plane partitions and their generating functions. This talk is based on joint work with Gregg Musiker, Nick Ovenhouse, and Ralf Schiffler.