Work

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

Year: 2025

Type: article

Abstract: Let V and W be quiver representations over $$\mathbb {F}_1$$ F 1 and let K be a field. The scalar extensions $$V^K$$ V K and $$W^K$$ W K are quiver representations over K with a distinguished, very well-behaved basis. We construct a basis of $${{\,\textrm{Hom}\,}}_{KQ}(V^K,W^K)$$ 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.

Source: Algebras and Representation Theory

DOI: 10.1007/s10468-025-10326-9

Author: Markus Kleinau

Institution: University of Bonn

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,490

Funder: Rheinische Friedrich-Wilhelms-Universität Bonn