Symplectic Grassmannians and Cyclic Quivers

TL;DR

This paper extends quiver Grassmannian theory to symplectic cases, constructing cellular decompositions, analyzing automorphisms, and embedding into affine flag varieties, thus advancing the understanding of symplectic algebraic varieties.

Contribution

It introduces symplectic quiver Grassmannians, develops their combinatorics, and connects them to classical symplectic Grassmannians and affine flag varieties.

Findings

Constructed cellular decompositions of symplectic quiver Grassmannians.

Number of irreducible components equals the Euler characteristic of classical symplectic Grassmannians.

Described automorphism groups and orbit structures of the varieties.

Abstract

The goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincar\'e polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.