Steinitz's problem asks whether a triangulated sphere is realizable geometrically as the boundary of a convex polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their creation 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 an approach to study the problem grouping together Schur functions, combinatorics of words, and oriented matroids.
Universal oriented matroids for subword complexes of Coxeter groups
JeanPhilippe Labbé, Freie Universität Berlin

PDL C401