Séminaires : Séminaire d'Algèbre

One of the most celebrated theorems in the theory of quiver representations is undoubtedly Gabriel’s theorem, which reveals a deep connection between quiver representations and root systems arising from Lie algebras. In particular, Gabriel’s theorem shows that the finite-type quivers are classified by the ADE Dynkin diagrams and the indecomposable representations are in bijection with the underlying positive roots. Following the works of Dlab—Ringel, the classification can be generalised to include all the other Dynkin diagrams (including BCFG) if one considers the more general notion of valued quivers (K-species) representations instead.

While the theories above relate (valued) quiver representations to root systems arising from Lie algebras, the aim of this talk is to generalise Gabriel’s theorem in a slightly different direction using root systems arising in Coxeter theory. Namely, we shall introduce a new notion of Coxeter quivers and their representations built in (other) fusion categories, where we have a generalised Gabriel’s theorem as follows: a Coxeter quiver has finitely many indecomposable representations if and only if its underlying graph is a Coxeter-Dynkin diagram — including the non-crystallographic types H and I. Using a similar notion of reflection functors as introduced by Bernstein—Gelfand—Ponomarev, we shall also show that the isomorphism classes of indecomposable representations of a Coxeter quiver are in bijection with the positive roots associated to the root system of the underlying Coxeter graph. --

This talk will take place in hybrid mode at the IHP.