Scalar extensions of quiver representations over F₁

Let V and W be quiver representations over F₁ and let K be a field. The scalar extensions VK and WK are quiver representations over K with a distinguished, very well-behaved basis. We construct a basis of Hom₊ₐ (VK, WK) 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.