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- Title
On Quiver Representations over F1.
- Authors
Jun, Jaiung; Sistko, Alexander
- Abstract
We study the category Rep (Q , F 1) of representations of a quiver Q over "the field with one element", denoted by F 1 , and the Hall algebra of Rep (Q , F 1) . Representations of Q over F 1 often reflect combinatorics of those over F q , but show some subtleties - for example, we prove that a connected quiver Q is of finite representation type over F 1 if and only if Q is a tree. Then, to each representation V of Q over F 1 we associate a coefficient quiver Γ V possessing the same information as V . This allows us to translate representations over F 1 purely in terms of combinatorics of associated coefficient quivers. We also explore the growth of indecomposable representations of Q over F 1 - there are also similarities to representations over a field, but with some subtle differences. Finally, we link the Hall algebra of the category of nilpotent representations of an n-loop quiver over F 1 with the Hopf algebra of skew shapes introduced by Szczesny.
- Publication
Algebras & Representation Theory, 2023, Vol 26, Issue 1, p207
- ISSN
1386-923X
- Publication type
Article
- DOI
10.1007/s10468-021-10092-4