Valuated Delta Matroids and Principal Minors of Hermitian matrices

TL;DR

This paper introduces valuated delta-matroids, generalizing valuated matroids and delta-matroids, and demonstrates their connection to principal minors of Hermitian matrices over valued fields, revealing new structural properties.

Contribution

The paper defines valuated delta-matroids and establishes their properties and relation to Hermitian matrix principal minors, extending the theory of matroid valuations.

Findings

Valuated delta-matroids exhibit properties similar to valuated matroids.

They arise as valuations of principal minors of Hermitian matrices.

This generalizes previous notions of delta-matroid representability.

Abstract

In this paper we introduce valuated $\Delta$-matroids, a natural generalization of two objects of study in matroid theory: valuated matroids and $\Delta$-matroids. We show that these objects exhibit nice properties analogous to ordinary valuated matroids. We also show that these objects arise as the valuations of principal minors of a Hermitian matrix over a valued field, generalizing other forms of $\Delta$-matroid representability.