From dimers to webs

Abstract

We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of SL r -webs, and is built upon the r-fold dimer model on the network. When r equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When r equals 2 or 3, it is a reformulation of the SL 2 -and SL 3 -web immanants defined by the second author. The basic result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of SL r -webs, reproving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity.

Combinatorics of the double-dimer model

“…Kuo's theorem then follows from a Plücker relation. The connection between dimers and the totally nonnegative Grassmannian is detailed in [6,18]. Theorem 1.1.1 can also be proven using the Desnanot-Jacobi identity, which is also called Dodgson condensation.…”

“…Consequently, each double-dimer configuration is associated with a planar pairing of the nodes. For example, in Figure 2.1, the pairing of the nodes is ( (1,8), (3,4), (5,2), (7,6)).…”

“…For example, the normalized probability Pr that a random double-dimer configuration on six nodes has pairing ((1, 2), (3,4), (5,6)) is Pr…”

Braid group symmetries of Grassmannian cluster algebras

“…We note that, at least in the most naive way, the invariant [T ] ∈ C[ Gr(4, n)] represented by a tensor diagram T is only well-defined up to a sign. For brevity's sake, we will forgo a careful discussion of these signs, referring the reader to [8] or [23] for some possible choices of conventions. We illustrate the smallest non-Plücker SL 4 webs (both of which are cluster variables), called "octapods": (43) Using the SL 4 skein relations, one can write any element of the skein algebra as a linear combination of planar diagrams without 0-cycles or 2-cycles (once we have removed 2cycles, our double bond drawings are unambiguous).…”

Channels, Billiards, and Perfect Matching 2-Divisibility

“…Given a graph G, a perfect matching of G is a set of edges µ such that each vertex of G is contained in a unique edge in µ. We let m G denote the number of distinct perfect matchings of G. The problem of determining m G arises in various mathematical contexts, particularly in tiling problems, but also in statistical mechanics [6], spectral graph theory [8], network analysis [4], total positivity [11], and representation theory [5]. Exact formulas for m G over an infinite family of graphs are quite rare.…”