On the bounded derived category of $\mathsf{IGr}(3, 7)$

TL;DR

This paper constructs a minimal Lefschetz decomposition and establishes a full exceptional collection of equivariant vector bundles for the derived category of the isotropic Grassmannian IGr(3,7), advancing understanding of its categorical structure.

Contribution

It provides the first explicit minimal Lefschetz decomposition and a full exceptional collection for the derived category of IGr(3,7), a significant step in geometric representation theory.

Findings

Constructed a minimal Lefschetz decomposition of D^b(IGr(3,7))

Proved the existence of a full exceptional collection of equivariant vector bundles

Enhanced understanding of the categorical structure of isotropic Grassmannians

Abstract

We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\mathsf{IGr}(3,7)$. Moreover, we show that $\mathsf{IGr}(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles.