Electrical varieties as vertex integrable statistical models

Abstract

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain local Yang–Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks. We show that, in the important case of an electrical network on the standard graph introduced in [Curtis E B et al 1998 Linear Algebr. Appl. 283 115–50], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed in [Berenstein A et al 1996 Adv. Math. 122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory 9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra.

Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation

“…The second principal result is the reveal of the tetrahedral symmetry of the multivariate Tutte polynomial at the point n = 2. Therefore, we have a connection between the multivariate Tutte polynomial, functions on Lustig cluster manifolds [4] and its electrical analogues [12,19]. We would like to interpret this property as the critical point of the model described by the multivariate Tutte polynomial, and the tetrahedral symmetry as a longstanding analog of the conformal symmetry of the Ising model at the critical point [8].…”

“…We should mention the relation of this subject with the theory of cluster algebras. We suppose that the multivariate Tutte polynomial on standard graphs at the critical point n = 2 corresponds to the orthogonal version of the Lusztig variety [4] in the case of the unipotent group and the electrical variety [12] for the symplectic group.…”

Vertex electrical model: lagrangian and non-negative properties

“…It is a matrix which depends on at most n(n − 1)/2 parameters, these parameters are the conductivities of the edges of the network. According to [9] such a matrix belongs to the symplectic group Sp(n).…”

“…This solution is related to the so-called electrical solution of the Zamolodchikov tetrahedron equation [7]. This representation was found in Sergeyev's work [8] and led to the modern understanding of electrical manifolds proposed in the work [9].…”

“…One of the main results from [9] is the following Lemma 2.1. For an electrical network e ∈ E n on the standard critical graph Σ n the row spaces of the matrices…”