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

TL;DR

This paper explores scalar extensions of quiver representations over the field with one element, providing a basis for morphism spaces and characterising absolutely indecomposable representations, thus advancing understanding in representation theory over $\\mathbb{F}_1$.

Contribution

It introduces a basis for morphism spaces in scalar extensions over $\\mathbb{F}_1$ and characterizes absolutely indecomposable representations, answering an open question.

Findings

Constructed a basis for morphism spaces between scalar extended representations.

Provided a combinatorial characterization of absolutely indecomposable representations.

Showed that indecomposables with finite nice length are absolutely indecomposable.

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.