2020/08/25 by Jaiung Jun, Jun, Jaiung, Alex Sistko +1 · 1 citation
Mathematics · #16G60 #17B35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 16G20 #Representation Theory (math.RT) #Secondary 05E10
paper · pdf · doi:10.48550/arxiv.2008.11304
openalex publication_date 2020/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the category \textrmRep(Q,\mathbbF1) of representations of a quiver Q over "the field with one element", denoted by \mathbbF1, and the Hall algebra of \textrmRep(Q,\mathbbF1). Representations of Q over \mathbbF1 often reflect combinatorics of those over \mathbbFq, but show some subtleties - for example, we prove that a connected quiver Q is of finite representation type over \mathbbF1 if and only if Q is a tree. Then, to each representation \mathbbV of Q over \mathbbF1 we associate a coefficient quiver Γ_\mathbbV possessing the same information as \mathbbV. This allows us to translate representations over \mathbbF1 purely in terms of combinatorics of associated coefficient quivers. We also explore the growth of indecomposable representations of Q over \mathbbF1 - 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 \mathbbF1 with the Hopf algebra of skew shapes introduced by Szczesny.