## Bergeron, Nantel and Ceballos, Cesar and Labbé, Jean-Philippe - Fan realizations of type $A$ subword complexes and multi-associahedra of rank 3

dmtcs:2512 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Fan realizations of type $A$ subword complexes and multi-associahedra of rank 3

Authors: Bergeron, Nantel and Ceballos, Cesar and Labbé, Jean-Philippe

We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$.

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Submitted on: November 21, 2016
Keywords: simplicial fans,multiassociahedra,subword complex,Gale duality,realization space,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]