On associahedra and related topics

Associahedra, Tamari Lattices and Related Structures, 2012

There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe threeconcerning special realizations with the vertices on a sphere, the space of all possible realizations, and possible realizations of the multiassociahedron.

2011

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.

Journal of Algebraic Combinatorics, 2014

We introduce, for any finite Coxeter group and nonnegative integer k, a spherical subword complex called the multi-cluster complex. For k = 1, this subword complex is isomorphic to the cluster complex of the given type. In particular, this result gives 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 done using only Coxeter groups theory. In types A and B, the presented complex coincides with known simplicial complexes, namely with the multi-associahedron of the given type. Moreover, 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 this particular simplicial complex.

Advances in Mathematics, 2015

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.

We show that three systematic construction methods for the n-dimensional associahedron,

2011

Given a finite Coxeter system (W, S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the c-generalized associahedron.

Proceedings of the American Mathematical Society, 2015

We show that the vertex barycenter of generalized associahedra and permutahedra coincide for any finite Coxeter system. Theorem 1.1. Let (W, S) be a finite Coxeter system, let c ∈ W be a Coxeter element, and let Asso u c (W ) be the c-associahedron obtained from a fairly balanced W -permutahedron Perm u (W ) by removing all boundary hyperplanes not containing a c-singleton. Then the vertex barycenters of Asso u c (W ) and Perm u (W ) coincide.

2014

Citation for published item: de Lucas, Miguel and Pu, Li and Turco, Gina Marie and Gaudinier, Allie and Marao, Ana Karina and Hiroshima, Hirofumi and Kim, Dahae and Ron, Mily and Sugimoto, Keiko and Roudier, Francois M. and Brady, Siobhan M. (2016) 'Transcriptional regulation of Arabidopsis Polycomb Repressive Complex 2 coordinates cell type proliferation and di erentiation.', Plant cell., 28 (10). pp. 2616-2631.

Discrete Mathematics, 2019

In the early 1990s, a family of combinatorial CW-complexes named permutoassociahedra was introduced by Kapranov, and it was realised by Reiner and Ziegler as a family of convex polytopes. The polytopes in this family are "hybrids" of permutohedra and associahedra. Since permutohedra and associahedra are simple, it is natural to search for a family of simple permutoassociahedra, which is still adequate for a topological proof of Mac Lane's coherence. This paper presents such a family.