On the Combinatorics of \mathbbF₁-Representations of Pseudotree Quivers

2023/01/17 by Jaiung Jun, Jaehoon Kim, Jun, Jaiung +3
Mathematics · #16G20 (Primary) 05E10 16G60 17B35 (Secondary) #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2301.07221

openalex publication_date 2023/01/17 · openalex created_date 2023/01/20 · openalex updated_date 2026/07/28

Abstract

We investigate quiver representations over \mathbbF₁. Coefficient quivers are combinatorial gadgets equivalent to \mathbbF₁-representations of quivers. We focus on the case when the quiver Q is a pseudotree. For such quivers, we first use the notion of coefficient quivers to provide a complete classification of asymptotic behaviors of indecomposable representations over \mathbbF₁. Then, we prove some fundamental structural results about the Lie algebras associated to pseudotrees. Finally, we construct examples of \mathbbF₁-representations M of a quiver Q by using coverings, under which the Euler characteristics of the quiver Grassmannians \textrmGr^Q_\underlined(M) can be computed in a purely combinatorial way.

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