On powers of Pl\"ucker coordinates and representability of arithmetic matroids

Abstract

The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Pl\"ucker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a vector of Pl\"ucker coordinates? For $k\neq 1$, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the subvariety of the Grassmannian that consists of all the points that define a regular matroid. The second topic is a related problem for arithmetic matroids. Let $\mathcal{A} = (E, rk, m)$ be an arithmetic matroid and let $k\neq 1 $ be a non-negative integer. We prove that if $\mathcal{A}$ is representable and the underlying matroid is non-regular, then $\mathcal{A}^k := (E, rk, m^k)$ is not representable. This provides a large class of examples of arithmetic matroids that are not representable. On the other hand, if the underlying matroid is regular and an additional condition is satisfied, then $\mathcal{A}^k$ is representable. Bajo-Burdick-Chmutov have recently discovered that arithmetic matroids of type $\mathcal{A}^2$ arise naturally in the study of colourings and flows on CW complexes. In the last section, we prove a family of necessary conditions for representability of arithmetic matroids.