Abstract: 

To any finite quiver Q, we associate a proto-exact category of nilpotent representations over F1, the so-called "field with one element." This category acts as a combinatorial analogue to the category of nilpotent k-representations of Q, where k is an arbitrary field. As in the case where k is a finite field, the category of nilpotent F1-representations of Q admits a well-defined Hall algebra. In this talk, we discuss recent progress towards understanding the combinatorial and algebraic structures of these categories and their associated Hall algebras. We focus particularly on the class of pseudotree quivers, which feature prominently in the resolution of an effort to stratify quivers by the complexity of their F1-representations. This is joint work with Drs. Jaiung Jun (SUNY New Paltz) and Jaehoon Kim (KAIST).