Total positivity for the Lagrangian Grassmannian

The stratification of the Grassmannian by positroid varieties has been the subject of extensive research. Positroid varieties are in bijection with a number of combinatorial objects, including $k$-Bruhat intervals and bounded affine permutations. In addition, Postnikov's boundary measurement map gives a family of parametrizations of each positroid variety; the domain of each parametrization is the space of edge weights of a weighted planar network. In this paper, we generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian $\Lambda(2n)$, which is the type $C$ analog of the ordinary, or type $A$, Grassmannian. The Lagrangian Grassmannian has a stratification by projected Richardson varieties, which are the type $C$ analogs of positroid varieties. We define type $C$ generalizations of bounded affine permutations and $k$-Bruhat intervals, as well as several other combinatorial posets which index positroid varieties. In addition, we generalize Postnikov's network parametrizations to projected Richardson varieties in $\Lambda(2n)$. In particular, we show that restricting the edge weights of our networks to $\mathbb{R}^+$ yields a family of parametrizations for totally nonnegative cells in $\Lambda(2n)$. In the process, we obtain a set of linear relations among the Pl\"ucker coordinates on $\text{Gr}(n,2n)$ which cut out the Lagrangian Grassmannian set-theoretically.