Permutahedra and generalized associahedra

Details Permutahedra and generalized associahedra Permutahedra and generalized associahedra

2007-09-26

arxiv.org/abs/0709.4241v3

Advances in Math., 226 (2011), pp.608-640

Mathematics

Combinatorics

27 pages, 10 figures; v3: 31 pages, 10 figures, Section 3 is rewritten, corrected typos, and updated references

Scientific paper

Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the cones in the $c$-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if $W$ is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to $W$, then we show that our realizations have integer coordinates in this lattice.

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