A matroidal perspective on the tropical Prym variety

We associate a matroid $M(\tilde{\mathrm{\Gamma }}\mathrm{/}\mathrm{\Gamma })$ to a harmonic double cover $\pi :\tilde{\mathrm{\Gamma }}\to \mathrm{\Gamma }$ of metric graphs. The matroid $M(\tilde{\mathrm{\Gamma }}\mathrm{/}\mathrm{\Gamma })$ is a geometric interpretation of Zaslavsky’s signed graphic matroid. We show that the principalization ${\mathrm{P}\mathrm{r}\mathrm{y}\mathrm{m}}_p(\tilde{\mathrm{\Gamma }}\mathrm{/}\mathrm{\Gamma })$ of the tropical Prym variety of the double cover can be reconstructed from $M(\tilde{\mathrm{\Gamma }}\mathrm{/}\mathrm{\Gamma })$, equipped with certain additional decorations. We describe the simplification of the matroid $M(\tilde{\mathrm{\Gamma }}\mathrm{/}\mathrm{\Gamma })$ and show that the Prym variety does not change under simplification.