F1 Representations and Hall Algebras

For any quiver Q, one can associate a category of Q-representations over F1, the so-called “field with one element.” This category, and its associated Ringel-Hall algebra, retain many features of representations over fields while exhibiting interesting differences. In this talk, we discuss recent advances in the study of F1-representations and their Hall algebras. After an overview of the fundamental background, we describe how F1-representations may be studied via coefficient quivers. This approach yields results on representation type over F1 and new insights into the associated Hall algebras. With the remaining time, we discuss an ongoing project which applies F1-representation theory to compute the Euler characteristics of certain quiver Grassmannians. This is joint work with Jaiung Jun.