Some Lagrangian quiver Grassmannians for the equioriented cycle
Matteo Micheli
TL;DR
The paper analyzes the symplectic degeneration $X(n,2n)^{sp}$ of the Lagrangian Grassmannian via a quiver Grassmannian framework, establishing a robust combinatorial model using symplectic juggling patterns and symplectic bounded affine permutations. It proves that $X(n,2n)^{sp}$ is a GKM variety under the subtorus $T^{sp}$, and that the orbit closures correspond exactly to a symplectic Bruhat order in affine type $C$, with orbit dimensions given by the symplectic length $oldsymbol{ ext{ell}}^{sp}(f_{oldsymbol{ ext{J}}})$. The results show that symplectic mutations encode elementary Bruhat relations, ensuring that closure relations can be read off from 1-dimensional $T^{sp}$-orbits, and provide a precise dimension formula for the symplectic cells. A small-case equivariant cohomology calculation is presented in the Appendix, illustrating the concrete computational framework. Collectively, the work advances understanding of degenerations of isotropic Grassmannians by linking geometric stratifications to an affine Coxeter combinatorics of type $C$.
Abstract
The goal of this paper is to better understand a family of linear degenerations of the classical Lagrangian Grassmannians $Λ(2n)$. 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 Pütz. 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 $Λ(2n)$ 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.