Department of Mathematics Colloquium


Abstract 

A hyperplane arrangement is a union of
hyperplanes in an vector space. One of the primary examples is
the braid arrangement, which is cut out by the root system of GL(n).
The flag variety is an algebraic variety associated to GL(n)
which parametrizes full flags in a vector space. A classic theorem
states that the topology of the flag variety and the topology of the
braid arrangement are numerically connected by certain integers which
show up in the representation theory of the symmetric group. Oh,
Postnikov, and Yoo have shown that this picture extends to any smooth
Schubert variety, with the associated inversion arrangement taking the
place of the braid arrangement. In this talk, I will show that Oh, Postnikov, and Yoo's theorem extends to rationally smooth Schubert varieties of arbitrary type. Furthermore, with a few additional related conditions (in particular, freeness of the inversion arrangement) this picture characterizes the rationally smooth Schubert varieties. I will also show that a simple combinatorial procedure, Peterson translation, can be used to determine when an arbitrary inversion arrangement is free. A large portion of the talk should be understandable with only elementary group theory and linear algebra.
