Subword complexes, cluster complexes, and generalized multi-associahedra

Details Subword complexes, cluster complexes, and generalized multi-associahedra Subword complexes, cluster complexes, and generalized multi-associahedra

2011-08-08

arxiv.org/abs/1108.1776v3

Mathematics

Combinatorics

24 pages, corrected many typos, two new sections, 3 Tables, 2 Figures

Scientific paper

We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, this subword complex is isomorphic to the cluster complex of the given type. In particular, we obtain a simple combinatorial description of the compatibility relation among almost positive roots. This approach generalizes results by K. Igusa and R. Schiffler in crystallographic types, and is developed purely in the context of Coxeter group theory. We show that in types A and B, the introduced complexes coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations respectively. We give an alternative definition of multi-cluster complexes in terms of the strong intervening neighbors property, and we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex. Moreover, we describe a natural cyclic action on multi-cluster complexes that yields a connection between multi-cluster complexes, Auslander-Reiten quivers and repetition quivers. Finally, we discuss further directions and present several conjectures.

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