Primary ideals associated to the linear strands of Lascoux's resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra gl(m|n)

We investigate equivariant Koszul duality between primary ideals Ia×b of S=S(M∗(m×n)) associated to rectangular Young diagrams a×b and the corresponding atypical irreducible mixed supertensor representations Xa×b of gl(m|n) in characteristic zero. We show Ia×b to be H0(S⊗Xa×b) of the Koszul dual S⊗Xa×b and we compute all the higher cohomology of S⊗Xa×b to be direct sums of I(a+r)×(b+r) with multiplicities being given by coefficients of Gauss polynomials. Utilizing this we are able to describe the equivariant syzygies of Xa×b over S!=Λ(M(m×n)) and determine the homological dimension of Ia×b over S.