Department of Mathematics Colloquium

University of Idaho

Fall 2013

Thursday,  September 12, 3:30-4:20 pm, room TLC 145

Refreshments in Brink 305 at 3:00 pm

Inversion hyperplane arrangements, Schubert varieties,
and Peterson translation


William Slofstra

Department of Mathematics

University of California, Davis


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.