Total Positivity and Network Parametrizations: From Type A to Type C.

Abstract

The Grassmannian Gr(k,n) of k-planes in n-space has a stratification by positroid varieties, which arises in the study of total nonnegativity. The positroid stratification has a rich combinatorial theory, introduced by Postnikov. In the first part of this thesis, we investigate the relationship between two families of coordinate charts, or parametrizations, of positroid varieties. One family comes from Postnikov's theory of planar networks, while the other is defined in terms of reduced words in the symmetric group. We show that these two families of parametrizations are essentially the same. In the second part of this thesis, we extend positroid combinatorics to the Lagrangian Grassmannian, a subvariety of Gr(n,2n) whose points correspond to maximal isotropic subspaces with respect to a symplectic form. Applying our results about parametrizations of positroid varieties, we construct network parametrizations for the analogs of positroid varieties in the Lagrangian Grassmannian using planar networks which satisfy a symmetry condition.