Representation Theory of Quivers Over the Field with One Element

This thesis studies representations of quivers over the field with one element, F1. First,we introduce F1-vector spaces, covering subspaces, quotient spaces, and linear maps. Wedescribe how F1-linear maps can be represented using binary matrices and explore keyproperties such as kernels, images, and cokernels. We also state Noether’s First IsomorphismTheorem in this setting.Next, we analyze the normal form of endomorphisms over F1. We study nilpotent maps,direct sum decompositions, and tensor products. Unlike in classical algebra, neither thedirect sum nor the tensor product satisfies a universal property. We then discuss Jordanblocks and their role in representing nilpotent and cyclic maps, leading to the Jordannormal form in the F1-framework.Finally, we focus on quiver representations over F1. We define basic concepts, morphisms,and direct sums of representations. We classify indecomposable representations and studyrepresentations of quivers of tree type. Equivalence relations on representations are introduced,with a special focus on those generated by a given relation. This work helps inunderstanding how classical representation theory changes when adapted to F1.