Lindstr\"om's conjecture on a class of algebraically non-representable matroids

TL;DR

This paper proves Lindström's conjecture that certain algebraically non-representable matroids, specifically harmonic matroids, are not algebraically representable when the parameter n is composite, extending previous results.

Contribution

The paper introduces harmonic matroids and proves Lindström's conjecture for this broader class, showing non-representability for composite n.

Findings

Harmonic matroids include full algebraic matroids as examples.

Lindström's conjecture holds for harmonic matroids when n is composite.

The result extends non-representability beyond previous cases.

Abstract

Gordon introduced a class of matroids $M(n)$, for prime $n\ge 2$, such that $M(n)$ is algebraically representable, but only in characteristic $n$. Lindstr\"om proved that $M(n)$ for general $n\ge 2$ is not algebraically representable if $n>2$ is an even number, and he conjectured that if $n$ is a composite number it is not algebraically representable. We introduce a new kind of matroid called {\it harmonic matroids}, of which full algebraic matroids are an example. We prove the conjecture in this more general case.