Primary ideals associated to the linear strands of Lascoux's resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra \(\mathfrak {gl}(m|n)\) (Q875913)

From the authors' abstract: ``We investigate equivariant Koszul duality between primary ideals \(I_{a\times b}\) of \(S=S(M(m\times n)^*)\) associated to rectangular Young diagrams \(a\times b\) and the corresponding atypical irreducible mixed supertensor representations \(X_{a\times b}\) of \(gl(m|n)\) in characteristic zero. We show \(I_{a\times b}\) to be \(H^0(S\otimes X_{a\times b})\) of the Koszul dual \(S\otimes X_{a\times b}\) and we compute all the higher cohomology of \(S\otimes X_{a\times b}\) to be direct sums of \(I_{(a+r)\times(b+r)}\) with multiplicities being given by coefficients of Gauss polynomials. Utilizing this we are able to describe the equivariant syzygies of \(X_{a\times b}\) over \(S^!=\bigwedge(M(m\times n))\) and determine the homological dimension of \(I_{a\times b}\) over \(S\).''