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Campus inMoscow

# Seminars "Combinatorics of invariants". Speaker: Grisha Taroyan

Event ended

Kasteleyn theorem, totally nonnegative Grassmannians and geometric signatures

Kasteleyn theorem [2] provides an elegant way to count the number of perfect match- ings (or dimer configurations) on a planar bipartite graph using a Kasteleyn matrix. This result can be refined for bipartite planar graphs in a disk [4] with weights. As it turns out this purely combinatorial subject is closely related to the topology of Postnikov’s totally nonnegative Grassmannians defined in [3]. Proposition ([4]). Let GG(G, I, w) be the number of perfect matchings with boundary condition I and weight w. Then the map w → (GG(G, I, w))I∈([n]\choose k) parametrizes a cell of totally nonnegative Grassmannian corresponding to the graph G. As it turns out two additional parametrizations of the same portion could be given. One of them uses the boundary measurement map of [3] and another one uses the minors of the weighted Kasteleyn matrix. In [1] it is shown that these parametrizations are equivalent. Furthermore, Kasteleyn signatures used in the definition of Kasteleyn matrices are geometric in nature, which allows for applications to solutions of KP-II PDE. In the first part of the talk we will recall a version of the Kasteleyn theorem due to [4] and some well-known results on topology of totally nonnegative Grassmannians. The second part of the talk will be an attempt to understand the geometry of Kasteleyn signatures and matrices following [1]. Finally, if time permits we will try to understand the significance of these geometric properties for solutions of the KP-II equation.

References

[1] Simonetta Abenda. “Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk”. In: Mathematical Physics, Analysis and Geometry 24.4 (2021), pp. 1–64.

[2] Pieter Kasteleyn. “Graph theory and crystal physics”. In: Graph theory and theoretical physics (1967), pp. 43–110.

[3] Alexander Postnikov. “Total positivity, Grassmannians, and networks”. In: arXiv preprint math/0609764 (2006). [4] David E Speyer. “Variations on a theme of Kasteleyn, with Application to the Totally Non- negative Grassmannian”. In: The Electronic Journal of Combinatorics (2016), P2–24.