The University of Michigan Combinatorics Seminar
We start from the following problem: given two opposite complete real n-dimensional flags, find the number of connected components of the set of real flags intersecting with the two initial flags transversally. Direct calculation gives values 2,6,20,52 for n=2,3,4,5. Oddly enough, this irregular sequence is extended by 3·2n-1 for n>5. The proof follows from a bijection between the connected components in question and the orbits of a certain linear group generated by symplectic transvections. This result can be further generalised to enumerating connected components in real double Bruhat cells for semisimple simply connected Lie groups. Going even further, we arrive at counting components of the union of generic symplectic leaves of a cluster manifold, where a similar result holds true.
This talk is based on joint work with M.Gekhtman (Notre Dame U.), B.Shapiro (Stockholm U.), M.Shapiro (Michigan State U.), and A.Zelevinsky (Northeastern U.).