Valuated Delta Matroids and Principal Minors of Hermitian matrices

In this paper we introduce valuated $\mathrm{\Delta }$-matroids, a natural generalization of two objects of study in matroid theory: valuated matroids and $\mathrm{\Delta }$-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 $\mathrm{\Delta }$-matroid representability.