Tropical Quiver Representations

Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Quiver Grassmannians are generalizations of these spaces arise in representation theory as the moduli spaces of quiver subrepresentations. These represent arrangements of vector subspaces satisfying linear relations provided by a directed graph.

Using tropical geometry, we introduce matroidal and tropical analoga of quivers and their Grassmannians obtained in joint work with Alessio Borzì and separate joint work with Giulia Iezzi; and describe them as affine morphisms of valuated matroids and linear maps of tropical linear spaces. Further, I will discuss first steps towards a polyhedral characterization of morphisms, and give an overview of recent work in progress on applying and extending the theory.