Permutahedra and generalized associahedra. (English) Zbl 1233.20035

Hohlweg, Christophe; Lange, Carsten E. M. C.; Thomas, Hugh

Adv. Math. 226, No. 1, 608-640 (2011).

Suppose that \((W,S)\) is a finite Coxeter system and \(c\in W\) is a Coxeter element (a product of the simple reflections in some order). N. Reading [Adv. Math. 205, No. 2, 313-353 (2006; Zbl 1106.20033)] has associated to this data the corresponding \(c\)-Cambrian fan \(\mathcal F_c\) and conjectured that every such fan is the normal fan of a polytope.
The main result of this article is a proof of this conjecture, via the construction of a \(c\)-generalized associahedron. Note that Cambrian fans are known to be linearly isomorphic to \(\mathbf g\)-vector fans of finite type cluster algebras with respect to an acyclic initial seed. This was proved by N. Reading and D. E. Speyer [in J. Eur. Math. Soc. 11, No. 2, 407-447 (2009; Zbl 1213.20038)] modulo a conjecture of S. Fomin and A. Zelevinsky [Compos. Math. 143, No. 1, 112-164 (2007; Zbl 1127.16023)] which was later proved by S.-W. Yang and A. Zelevinsky [Transform. Groups 13, No. 3-4, 855-895 (2008; Zbl 1177.16010)].

Reviewer: Robert Marsh (Leeds)

MSC:

20F55	Reflection and Coxeter groups (group-theoretic aspects)
52B11	\(n\)-dimensional polytopes
05E15	Combinatorial aspects of groups and algebras (MSC2010)
06A07	Combinatorics of partially ordered sets

Keywords:

Coxeter groups; Cambrian fans; Cambrian lattices; generalized associahedra; cluster fans; Weyl groups; cluster algebras; Bruhat order; Coxeter singleton

Software:

GAP; CHEVIE

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