On Quiver Representations over $\mathbb {F}_{1}$

Abstract

We study the category \(\text {Rep}(Q,\mathbb {F}_{1})\) of representations of a quiver Q over “the field with one element”, denoted by \(\mathbb {F}_{1}\), and the Hall algebra of \(\text {Rep}(Q,\mathbb {F}_{1})\). Representations of Q over \(\mathbb {F}_{1}\) often reflect combinatorics of those over \(\mathbb {F}_{q}\), but show some subtleties - for example, we prove that a connected quiver Q is of finite representation type over \(\mathbb {F}_{1}\) if and only if Q is a tree. Then, to each representation \(\mathbb {V}\) of Q over \(\mathbb {F}_{1}\) we associate a coefficient quiver \({\Gamma }_{\mathbb {V}}\) possessing the same information as \(\mathbb {V}\). This allows us to translate representations over \(\mathbb {F}_{1}\) purely in terms of combinatorics of associated coefficient quivers. We also explore the growth of indecomposable representations of Q over \(\mathbb {F}_{1}\) - there are also similarities to representations over a field, but with some subtle differences. Finally, we link the Hall algebra of the category of nilpotent representations of an n-loop quiver over \(\mathbb {F}_{1}\) with the Hopf algebra of skew shapes introduced by Szczesny.