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 \({{\,\textrm{Hom}\,}}_{KQ}(V^K,W^K)\) 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.