On powers of Plücker 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 Plücker 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 Plücker 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 nonnegative integer. We prove that if $\mathcal{A}$ is representable and the underlying matroid is nonregular, 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. BajoBurdickChmutov 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.10520
 Bibcode:
 2017arXiv170310520L
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry;
 Primary: 05B35;
 14M15;
 Secondary: 14T05
 EPrint:
 36 pages, 1 figure, minor corrections, same content as journal version