Constructing equivariant vector bundles via the BGG correspondence

We describe a strategy for the construction of finitely generated G-equivariant Z-graded modules M over the exterior algebra for a finite group G. By an equivariant version of the BGG correspondence, M defines an object F in the bounded derived category of G-equivariant coherent sheaves on projective space. We develop a necessary condition for F being isomorphic to a vector bundle that can be simply read off from the Hilbert series of M. Combining this necessary condition with the computation of finite excerpts of the cohomology table of F makes it possible to enlist a class of equivariant vector bundles on P4 that we call strongly determined in the case where G is the alternating group on 5 points.