Primary ideals associated to the linear strands of Lascouxʹs resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra gl(mn)

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(mn) 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.