Tropical Quiver Grassmannians

TL;DR

This paper introduces tropical parameter spaces for quivers of valuated matroids, defining quiver Dressians and exploring their properties, including realizability issues in low dimensions.

Contribution

It develops the theory of quiver Dressians and their relation to tropical linear spaces, extending the understanding of tropical Grassmannians and their realizability.

Findings

Quiver Dressians parametrize tropical linear space containment.

Tropicalizations of quiver Grassmannians correspond to realizable cases.

Nonrealizable points appear in quiver Dressians starting from dimension 2.

Abstract

We introduce quivers of valuated matroids and study their tropical parameter spaces. We define quiver Dressians, which parametrize containment of tropical linear spaces after tropical matrix multiplication, and show that tropicalizations of quiver Grassmannians parametrize the realizable analogue. We further introduce affine morphisms of valuated matroids and show compatibility with weakly monomial quiver representations. Finally, we show that starting in ambient dimension 2, quiver Dressians can have nonrealizable points.