Coxeter Elements and Periodic Auslander-Reiten Quiver

Details Coxeter Elements and Periodic Auslander-Reiten Quiver Coxeter Elements and Periodic Auslander-Reiten Quiver

2007-03-12

arxiv.org/abs/math/0703361v2

Mathematics

Representation Theory

27 pages, 10 figures. v2: Added new sections relating our results to the theory of quiver representations

Scientific paper

In this paper we show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of $R$ with a certain subset $Ihat$ of $I x Z_{2h}$, where $h$ is the Coxeter number. The set $Ihat$ has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram $I$: roots are vertices of $Ihat$, and the root lattice and the inner product admit an explicit description in terms of $Ihat$. Finally, we relate this construction to the theory of quiver representations.

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