Scalar extensions of quiver representations over ${\mathbb{F}}_1$

Abstract

Let $V$ and $W$ be quiver representations over ${\mathbb{F}}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of ${\mathrm{Hom}}_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.

17 pages, comments welcome, V2: Improvements to exposition based on referee feedback, to appear in Algebras and Representation Theory