Networks on surfaces and cells in Grassmannians

by Alek Vainshtein

We start from a construction, due to Postnikov, that parametrizes
cells in a Grassmannian by directed trivalent networks in a disk
(S^1 with one hole). We discuss equivalent transformations of these
networks, minimality conditions and orientation switches. Next, we
extend this theory to directed trivalent networks on a cylinder
(S^1 with two holes), thus providing a parametrization of polynomial loops
in Grassmannians. In the case of a disk, this parametrization naturally
leads to the Sklyanin Poisson-Lie structure on GL(n) and its push-forward to Grassmannians as Poisson homogeneous spaces, while in the case of a cylinder one recovers a Poisson bracket corresponding to the classical trigonometric R-matrix.

Based on joint work with M.Gekhtman (Notre Dame) and M.Shapiro (East
Lansing).