The University of Michigan Combinatorics Seminar


Abstract 

I will consider Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. This map turns out to be Poisson, provided the space of edge weights is equipped with a representative of a 6parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriately chosen Rmatrix PoissonLie structure. I will explain that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure. Finally, I will extend Postnikov's model to the case of networks embedded into an annulus, which leads to a map into the space of loops in the Grassmannian. Natural Poisson brackets on edge weights in this case are intimately connected to trigonometric Rmatrix brackets on matrixvalued rational functions. Based on joint work with M.Gekhtman and M.Shapiro. 