Electrical networks, Grassmannians, and cluster algebras

An electrical network with $n$ boundary vertices induces a matrix called the response matrix which measures the electrical properties of the network. The set of response matrices of all electrical networks has a characterization in terms of positivity of circular minors. Alman, Lian and Tran constructed a cluster algebra on the set of circular minors, which encodes the tests for positivity of these minors. Lam established the embedding of the set of electrical networks with $n$ boundary vertices into the totally nonnegative Grassmannian $Gr_{\ge0}(n-1,2n)$. The coordinate ring of the Grassmannian has a cluster algebra structure as was proved by Scott. Given an electrical network, we find a relation between circular minors of its response matrix and Plücker coordinates of its image in the Grassmannian. Using this property, we prove that for an odd $n$ the two cluster algebras, on circular minors and on the Grassmanian, become isomorphic after a natural freezing and subsequent trivialization of certain variables in their initial seeds. We apply this isomorphism in order to relate the tests for positivity of circular minors to tests for positivity in the Grassmannian.