Fan Realizations of Type $$A$$ A Subword Complexes and Multi-associahedra of Rank 3

Abstract

We present complete simplicial fan realizations of any spherical subword complex of type $$A_n$$ for $$n\le 3$$ . This provides complete simplicial fan realizations of simplicial multi-associahedra $$\varDelta _{2k+4,k}$$ , 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. We also present fan realizations of two previously unknown cases of subword complexes of type $$A_4$$ , namely the multi-associahedra $$\varDelta _{9,2}$$ and $$\varDelta _{11,3}$$ .