Generalized juggling patterns, quiver Grassmannians and affine flag varieties

TL;DR

This paper explores the connections between quiver Grassmannians, totally nonnegative Grassmannians, and local models of Shimura varieties, providing new geometric realizations and explicit embeddings into affine flag varieties.

Contribution

It generalizes previous work linking quiver Grassmannians to totally nonnegative Grassmannians and describes explicit embeddings into affine flag varieties for local models of Shimura varieties.

Findings

Explicit embeddings of quiver Grassmannians into affine flag varieties.

Realization of quiver Grassmannians as unions of Schubert varieties.

Generalization of combinatorial and geometric relations between these structures.

Abstract

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.