A Note on Extension Properties and Representations of Matroids

TL;DR

This paper explores extension properties of matroids and polymatroids, using computational methods to identify non-linearly representable matroids and analyzing specific classes like sparse paving matroids with the tic-tac-toe configuration.

Contribution

It introduces new extension property criteria for matroid representations and provides computationally discovered examples of non-linear representability, especially in sparse paving matroids.

Findings

Identified new non-linearly representable matroids through computer-aided exploration.

Analyzed extension properties of matroids on eight and nine elements.

Provided a clearer description of sparse paving matroids with the tic-tac-toe configuration.

Abstract

We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic representations. Iterations of those extension properties are checked for matroids on eight and nine elements by means of computer-aided explorations, finding in that way several new examples of non-linearly representable matroids. A special emphasis is made on sparse paving matroids on nine points containing the tic-tac-toe configuration. We present a new, more clear description of that family and we analyze extension properties on those matroids and their duals.