Master Degree Project Presentation: Representation Theory of Quivers Over the Field with One Element

Abstract: 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. We describe how F1-linear maps can be represented using binary matrices and explore key properties such as kernels, images, and cokernels. We also state Noether’s First Isomorphism Theorem 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 the direct sum nor the tensor product satisfies a universal property. We then discuss Jordan blocks and their role in representing nilpotent and cyclic maps, leading to the Jordan normal 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 study representations 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 in understanding how classical representation theory changes when adapted to F1.