Equivariant Ulrich bundles on exceptional homogeneous varieties

TL;DR

This paper classifies exceptional homogeneous varieties with Picard number 1 that admit irreducible equivariant Ulrich bundles, identifying only the Cayley plane and the E7-adjoint variety, and explores related bundle existence on hyperplane sections.

Contribution

It provides a complete classification of such varieties with irreducible equivariant Ulrich bundles among exceptional groups, highlighting the uniqueness of the Cayley plane and E7-adjoint variety.

Findings

Only the Cayley plane and E7-adjoint variety admit irreducible equivariant Ulrich bundles.

Hyperplane sections of the Cayley plane can have equivariant but non-irreducible Ulrich bundles.

The classification narrows the scope of varieties with these bundles in exceptional groups.

Abstract

We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane $E_6/P_1$ and the $E_7$-adjoint variety $E_7/P_1$. From this result, we see that a general hyperplane section $F_4/P_4$ of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.