Scalar Extensions of Quiver Representations Over $$\mathbb {F}_1$$

Abstract Let V and W be quiver representations over $$\mathbb {F}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and let K be a field. The scalar extensions $$V^K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>V</mml:mi> <mml:mi>K</mml:mi> </mml:msup> </mml:math> and $$W^K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mi>K</mml:mi> </mml:msup> </mml:math> are quiver representations over K with a distinguished, very well-behaved basis. We construct a basis of $${{\,\textrm{Hom}\,}}_{KQ}(V^K,W^K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>Hom</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mi>KQ</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mi>K</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mi>K</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.