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Generalized associahedra via quiver representations. (English) Zbl 1042.52007
The paper provides a conceptual interpretation in terms of quivers for generalized associahedra introduced in [S. Fomin and A. Zelevinsky, Ann. Math. (2) 158, No. 3, 977–1018 (2003; Zbl 1057.52003)], where a smooth complete simplicial fan \(\Delta(\Phi)\) in the ambient real vector space is associated to a finite (crystallographic) root system \(\Phi\). Both \(\Delta(\Phi)\) and the corresponding simple polytope are called generalized associahedra associated to \(\Phi\). The construction includes as special cases the Stasheff polytope for \(\Phi\) of type \(A\) and the Bott-Taubes cyclohedron for \(\Phi\) of type \(B\) or \(C\). For a simply-laced root system \(\Phi\) corresponding to a Dynkin quiver \(\Gamma\), the authors establish a bijection between the almost positive roots in \(\Phi\) and the isoclasses of indecomposable decorated representations of \(\Gamma\), after a suitable modification of the dimension vectors. Further generalization of the reflection functors \(S_i\) and some other objects to the decorated case yield the desired interpretation. With this interpretation, the main results of S. Fomin and A. Zelevinsky [loc. cit.] are easy consequences of the facts about tilting representations [D. Happel and C. M. Ringel, Trans. Am. Math. Soc. 274, 399–445 (1982; Zbl 0503.16024)].

MSC:
52B11 \(n\)-dimensional polytopes
16G20 Representations of quivers and partially ordered sets
17B20 Simple, semisimple, reductive (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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