REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: A theorem of Ryan and Wolper states that every smooth Schubert variety of type A is an iterated fibre bundle of Grassmannians. I will talk about joint work with Ed Richmond extending this theorem to all finite types. In particular, Ryan and Wolper's theorem is closely related to the combinatorial notion of a Billey-Postnikov decomposition in the Weyl group. Using existence results for Billey-Postnikov decompositions, we also determine all (rationally) smooth Grassmannian Schubert varieties in every type, and apply these results to count the number of rationally smooth Schubert varieties in a full flag variety. Time permitting, I will discuss two important questions for Coxeter groups: when does an element have a Billey-Postnikov decomposition, and what type of Billey-Postnikov decompositions exist?