Equivariant Ulrich bundles on flag varieties

TL;DR

This paper investigates equivariant Ulrich bundles on flag varieties, classifies their existence on certain two-step varieties, and provides counterexamples to previous conjectures, advancing understanding of vector bundles in algebraic geometry.

Contribution

It classifies equivariant Ulrich bundles on specific flag varieties and offers a conjectural framework for their existence, challenging prior conjectures.

Findings

Partial flag varieties with three or more steps do not admit such Ulrich bundles.

Classified Ulrich bundles on certain two-step flag varieties.

Provided counterexamples to existing conjectures.

Abstract

In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(2,n;n+1), F(2,n;n+2), F(k,k+1;n), and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles. Our results provide counterexamples to conjectures by Costa and Miro-Roig.