College of Liberal Arts & Sciences

# Algebra Seminar

**Abstract: **

To any quiver, we can associate its category of finite-dimensional (nilpotent) representations over the field with one element. This category shares many basic properties with its analog over a field: in particular, a version of the Krull-Schmidt Theorem is satisfied. Inspired by the classical Tame-Wild Dichotomy for finite-dimensional algebras, we discuss a stratification of quivers based on the growth of their indecomposable F1-representations. In particular, we classify all quivers of bounded representation type over F1 and provide a functorial interpretation for unbounded quivers. As a consequence, we develop a general framework for interpreting F1-representations as certain quiver maps, which allows for a more combinatorial description of the Ringel-Hall algebras associated to these categories.