Some Lagrangian quiver Grassmannians for the equioriented cycle

TL;DR

This paper studies special degenerations of Lagrangian Grassmannians via quiver Grassmannians, proving conjectures about orbit orders, computing dimensions through affine permutations, and establishing GKM properties for these varieties.

Contribution

It proves a conjecture relating orbit closure order to combinatorial juggling patterns and shows that certain degenerations are GKM varieties.

Findings

Orbit order coincides with a combinatorial order on juggling patterns.

Orbit dimensions match lengths of affine permutations in type C Coxeter groups.

The varieties are GKM with respect to a subtorus action.

Abstract

The goal of this paper is to better understand a family of linear degenerations of the classical Lagrangian Grassmannians $\Lambda(2n)$. It is the special case for $k=n$ of the varieties $X(k,2n)^{sp}$, introduced in previous joint work with Evgeny Feigin, Martina Lanini and Alexander P\"utz. These varieties are obtained as isotropic subvarieties of a family of quiver Grassmannians $X(n,2n)$, and are acted on by a linear degeneration of the algebraic group $Sp_{2n}$. We prove a conjecture proposed in the paper above for this particular case, which states that the ordering on the set of orbits in $X(n,2n)^{sp}$ given by closure-inclusion coincides with a combinatorially defined order on what are called symplectic $(n,2n)$-juggling patterns, much in the same way that the $Sp_{2n}$ orbits in $\Lambda(2n)$ are parametrized by a type C Weyl group with the Bruhat order. The dimension of such orbits is computed via the combinatorics of bounded affine permutations, and it coincides with the length of some permutation in a Coxeter group of affine type C. Furthermore, the varieties $X(n,2n)$ are GKM, that is, they have trivial cohomology in odd degree and are equipped with the action of an algebraic torus with finitely many fixed points and 1-dimensional orbits. In this paper it is proven that $X(n,2n)^{sp}$ is also GKM, with respect to the action of a subtorus of the above torus.