Homogeneous Ulrich bundles on Flag manifolds

TL;DR

This paper investigates the existence and classification of irreducible homogeneous Ulrich bundles on various flag manifolds, providing explicit counts and non-existence results for specific cases.

Contribution

It classifies irreducible homogeneous Ulrich bundles on certain flag manifolds and establishes counts and non-existence results, proposing a conjecture for general cases.

Findings

All irreducible homogeneous Ulrich bundles on $f(0,n-1,n)$ are classified, with exactly $2^{n-1}$ such bundles.

Identifies specific flag manifolds supporting irreducible homogeneous Ulrich bundles, such as $f(0,n-2,n)$ and $f(1,n-1,n)$.

Proves the non-existence of such bundles on $f(0,1,n)$.

Abstract

Let $V$ be a $K$-vector space of dimension $n+1$. In this paper, we focus our attention into the existence of irreducible homogeneous Ulrich bundles on flag manifolds $\FF(p, q,n)$ which parameterizes all chains of linear subspaces $L_{p} \subset L_{q} \subset \PP(V)$ of dimension $p< q$, respectively. We determine all irreducible homogeneous Ulrich bundles on $\FF(0,n-1,n)$ and we prove that there are exactly $2^{n-1}$. Similarly, we prove that $\FF(0,n-2,n)$ and $\FF(1,n-1,n)$ are also the support of irreducible homogeneous Ulrich bundles. On the other hand, we prove that $\FF(0,1,n)$ do not support any irreducible homogeneous Ulrich bundle. We end posing a conjecture concerning the existence of irreducible homogeneous Ulrich bundles on $\FF(p,q,n)$ in terms of $p$ and $q$.