Total positivity for the Lagrangian Grassmannian

TL;DR

This paper extends the combinatorial and parametrization framework of positroid varieties from the ordinary Grassmannian to the Lagrangian Grassmannian, introducing type C analogs and total positivity structures.

Contribution

It introduces type C generalizations of positroid combinatorics and network parametrizations for the Lagrangian Grassmannian, expanding the understanding of its stratification and total positivity.

Findings

Defined type C bounded affine permutations and $k$-Bruhat intervals.

Generalized Postnikov's network parametrizations to the Lagrangian case.

Identified linear relations among Plücker coordinates for the Lagrangian Grassmannian.

Abstract

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.