Juggling patterns and the affine flag manifold

Motivated by study of "totally nonnegative matrices", Lusztig introduced
a natural stratification of partial flag manifolds that, in the
Grassmannian, was observed by Postnikov to have a surprising
cyclic symmetry. I'll describe this stratification, and explain how
to index the strata by "juggling patterns".

Mathematicians have studied these juggling patterns before, and
related them combinatorially to the affine Weyl group for GL(n).
This suggests that the cyclic symmetry of Lusztig et al's decomposition
has an affine origin, and indeed I'll derive it from the (much better
understood) Bruhat decomposition on the affine flag manifold.

This work is joint with Thomas Lam and David Speyer.