Fan realizations of subword complexes and multi-associahedra via Gale duality

TL;DR

This paper constructs complete simplicial fan realizations for certain subword complexes and multi-associahedra, solving open cases and providing methods applicable to all finite Coxeter groups.

Contribution

It introduces new fan realizations for spherical subword complexes of type A and multi-associahedra, including previously unknown cases, using Gale duality techniques.

Findings

Complete fan realizations for all spherical subword complexes of type A with n ≤ 3.

Fan realizations of multi-associahedra Δ_{2k+4,k} for various k.

New fan realizations of subword complexes of type A_4, specifically Δ_{9,2} and Δ_{11,3}.

Abstract

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 $\Delta_{11,3}$.