BGP-reflection functors and cluster combinatorics

Details BGP-reflection functors and cluster combinatorics BGP-reflection functors and cluster combinatorics

2005-11-15

arxiv.org/abs/math/0511380v3

Mathematics

Representation Theory

version 3

Scientific paper

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the "truncated simple reflections" on the set of almost positive roots $\Phi_{\ge -1}$ associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we give a unified interpretation via quiver representations for the generalized associahedra associated to the root systems of all Dynkin types (a simply-laced or non-simply-laced). This confirms the conjecture 9.1 in [4] in all Dynkin types.

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