Prerequisite

Basic knowledge of linear algebra, group theory and topology.

Introduction

An electrical network is a graph with positive weights attached to the edges and a chosen subset of the set of vertices which are called the boundary vertices. The theory of electrical network goes back to the work of Kirkhoff around mid 1800 and since then it has been a source of remarkable research in combinatorics, algebra, geometry and mathematical physics [7], [15], [16], [13], [14], [12], [9], [10].

In this course, we will describe some of these results and will build on them. First we will explain that there is a clear connection of the space of electrical networks $El$ with symplectic geometry. One can obtain a family of compactifications of $El$ in the Lagrangian Grassmannian $LG(n-1)$ and one can tailor the choice of an element in this family to a specific task. In particular, one can wish to parametrize the positive part of $LG(n-1)$ by electrical networks. We will construct such a parametrization which is in parallel with the work of Postnikov [17] on parametrization of the positive Grassmannian.

Such a parametrization is related to the electrical Lie group introduced by T. Lam and P. Pylyavskyy in [16], which is a partucular presentation of the simplectic group as they proved. Our second goal is to show that the notion of the electrical Lie group and Lie algebra can be generalized. In [3] the generalized electrical Lie algebra were defined for any given semisimple or Kac-Moody Lie algebra $\mathfrak{g}$ as a multiparametric family of Lie subalgebras of $\mathfrak{g}$ which are flat deformations of the nilpotent part $\mathfrak{n}$ of $\mathfrak{g}$.

Syllabus

1. Motivation: Totally positive matrices, positive Grassmannian, cluster algebras.

2. Electrical networks, the responce matrix. Combinatorial properties of the minors of the responce matrix.

3. The space of electrical networks: the star-triangle transformation, critical graphs, median graphs.

4. Electrical networks as an integrable system.

5. The space of electrical networks and the positive Lagrangian Grassmannian.

6. Electrical networks and the fundamental representation of the symplectic group.

7. Generalized electrical Lie groups and Lie algebras.

Reference

[1] Berenstein, A., Zelevinsky, A. Total positivity in Schubert varieties. Comment. Math. Helv. 72, 128–166 (1997). https://doi.org/10.1007/PL00000363

[2] Arkady Berenstein Sergey Fomin Andrei Zelevinsky, Parametrizations of Canonical Bases and Totally Positive Matrices, Advances in mathematics 122, 49-149 (1996).

[3] A. Berenstein, V. Gainutdinov, and V. Gorbounov, Generalized electrical Lie algebras, https://arxiv.org/abs/2405.02956 (2024).

[4] B. Bychkov, V. Gorbounov, A. Kazakov, D. Talalaev, Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups, Moscow Mathematical Journal, Vol.23, No.2, pp.133-167 (2023).

[5] Yibo Gao, Thomas Lam, Zixuan Xu, Electrical networks and the Grove algebra, Canadian journal of mathematics, pp.1-34 (2024).

[6] S. Chepuri, T. George and D. E. Speyer, Electrical networks and Lagrangian Grassmannians, https://arxiv.org/abs/2106.15418 (2021).

[7] E. B. Curtis, D. Ingerman, J. A. Morrow, Circular planar graphs and resistor networks. Linear algebra and its applications, Vol.283, No. 1-3, pp.115-150 (1998).

[8] V. Gorbounov and D. Talalaev, Electrical varieties as vertex integrable statistical models, arXiv:1905.03522 [math-ph]

[9] Kashaev, R.M., Korepanov, I.G. and Sergeev, S.M. Theor Math Phys (1998) 117: 1402. https://doi.org/10.1007/BF02557179

[10] S. Sergeev, Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-Electric Condition, Letters in Mathematical Physics 45(2), (1998), pp 113-119.

[11] R. Karpman, Total positivity for the lagrangian grassmannian, Adv. Appl. Math. 98, 25–76 (2018).

[12] R. W. Kenyon and D. B. Wilson, Boundary partitions in trees and dimers, Trans. Amer. Math. Soc., 363, pp. 1325–1364 (2011).

[13] R. Kenyon and D. Wilson, Combinatorics of tripartite boundary connections for trees and dimers. Electron. J Comb., 16(1) (2009).

[14] R. Kenyon and D. Wilson, The space of circular planar electrical networks, SIAM J. Discrete Math. 31(1), (2017) 1-28.

[15] T. Lam, Electroid varieties and a compactification of the space of electrical networks, Adv. Math. 338, pp. 549–600 (2018).

[16] T. Lam and P. Pylyavskyy, Electrical networks and Lie theory, Algebra and Number Theory, 9, pp.1401–1418 (2015).

[17] A. Postnikov, Total positivity, Grassmannians, and networks, https://arxiv.org/abs/math/0609764 (2006).

[18] Yi Su, Electrical Lie Algebra of Classical Types, Electrical Lie Algebra of Classical Types.

Lecturer Intro

Vasilii Gorbunov obtained a PhD in topology in 1987 from Novosibirsk State University. He held postdoc positions at University of Manchester, UK and Northwestern University, USA. After that he held Professor positions in University of Kentucky, USA and Aberdeen University, UK until 2020. Currently he is Professor at the Higher School of Economics in Moscow, Russia.