Symmetries on juggling varieties

This thesis aims to introduce and study an isotropic subvariety of a quiver Grassmannian for the equioriented cycle. That is, the subvariety of points consisting of collections of vector subspaces satisfying conditions derived from endowing the ambient space with a symplectic form. It seeks to merge and expand two articles that I have published during my PhD, one that I have co-authored and one under my sole name. It is the first step in this direction, as the symplectic approach has only ever been considered for acyclic quivers. Said quiver Grassmannian is obtained by linearly degenerating the action of the general linear group GL2n on the Grassmannian Gr(k, 2n) in a way that it is compatible with a fixed symplectic form on C 2n , so that the process restricts to both the symplectic group Sp2n and the isotropic Grassmannian Gr(k, 2n) sp. This results in the degeneration of the former acting on that of the latter, and producing a cellular decomposition of the variety into its orbits. We then investigate the poset structure on the set of orbits, given by inclusion of the closures. We explicitly describe the group, compute the dimension of the cells, and develop the underlying combinatorics. Particular focus is then given to the Lagrangian case, as it is especially interesting for dimensional reasons. Many more results can be proven in this case, such as describing the irreducible components of the subvariety, labeling each cell with an element of an affine Coxeter group of type C whose length equals the dimension of the cell, and equipping the subvariety with a skeletal action of an algebraic torus. This torus allows us to show that the closure-inclusion order on the set of orbits for the action of the symplectic subgroup on the subvariety is inherited by the closure-inclusion order on the set of orbits for the action of the ambient group on the ambient variety.