Electrical networks, Grassmannians, and cluster algebras

Abstract

The paper studies the problem of circular total positivity of the symmetric matrices with zero row sums. These matrices are exactly response matrices of the electrical networks. Alman, Lian and Tran described tests for circular total positivity in two related frameworks: the cluster algebra $\mathcal{CM}_n$ and the Laurent Phenomenon algebra $\mathcal{LM}_n$. Our first result is the construction of a seed in Scott's cluster algebra structure on the coordinate ring of the Grassmannian $\mathrm{Gr}(n-1,2n)$ that consists entirely of circular minors. We compare the cluster structure induced by this seed with $\mathcal{CM}_n$. In particular, for odd $n$ the cluster algebra structure $\mathcal{CM}_n$ is isomorphic to the cluster algebra structure on $\mathrm{Gr}(n-1,2n)$ subject to natural freezing and trivialization of certain cluster variables in their initial seeds. We use this isomorphism to relate circular total positivity to positivity in the Grassmannian. Our second result is that the Laurent Phenomenon algebra $\mathcal{LM}_n$ is isomorphic to the coordinate ring of the noncompactified space of electrical network, or equivalently, to a certain localization of the grove algebra.