Jean-Philippe Labbé (AG Diskrete Geometrie)

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.


Elise Walker (Texas A&M Univ.)


Katharina Jochemko (KTH Stockholm)


Andrés Vindas Meléndez (Univ. of Kentucky)


Thomas Yahl (Texas A&M Univ.)