  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.
