Electroid varieties: part one

In 2014, Lam showed that the space of electrical networks could be compactified and embedded as a linear slice of the totally nonnegative Grassmannian. This embedding gives rise to electroid varieties, defined to be the intersection of positroid varieties with the linear slice. Lam's original paper established numerous analogies between positroid varieties and electroid varieties. We build off of Lam's results and consider the algebro-geometric properties of electroid varieties, proving that they are irreducible, smooth, regular in codimension one, and compatibly Frobenius split.

This talk is based on joint work with Dawei Shen and David Speyer. In part one, I will define electroid varieties, introduce combinatorial tools for working with them, and prove that they are irreducible. Next week, Dawei will give a follow-up talk which will present results for the other algebro-geometric properties.