Harmonic Conjugation in Harmonic Matroids

TL;DR

This paper introduces harmonic matroids, a generalization of harmonic conjugation from geometry and algebraic matroids, enabling the construction of projective planes of prime order without traditional axioms.

Contribution

It develops the concept of harmonic conjugation within harmonic matroids and provides a combinatorial method to construct projective planes of prime order.

Findings

Harmonic conjugation can be extended to harmonic matroids.

Constructs a projective plane of prime order within harmonic matroids.

Provides a combinatorial construction in algebraic matroids.

Abstract

We study a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called \emph{harmonic matroids}. We use harmonic conjugation to construct a projective plane of prime order in harmonic matroids without using the axioms of projective geometry. As a particular case we have a combinatorial construction of a projective plane of prime order in full algebraic matroids.