Title: Combinatorial Foundations for Geometric Realizations of Subword Complexes of Coxeter Groups
Abstract: Let d>=1 and C be a simplicial complex homeomorphic to a (d-1)-dimensional sphere.
"Is there a d-dimensional simplicial convex polytope P whose face lattice is isomorphic to that of C?"
This is a classical problem in discrete geometry going back to Steinitz who proved that every 2-dimensional simplicial sphere is the boundary of a polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their introduction more than 15 years ago, subword complexes built up a wide portfolio of relations and applications to many other areas of research (Schubert varieties, cluster algebras, associahedra, tropical Grassmannians, to name a few) and a lot of efforts has been put into realizing them as polytopes, with relatively little success.
In this talk, I will present some reasons why this problem resisted so far, and present a glimpse of a novel approach to study the problem grouping togetherSchur functions, combinatorics of words, and oriented matroids.