Borel-Weil factorization for super Grassmannians

This article deals with computing the cohomology of Schur functors applied to tautological bundles on super Grassmannians. We show that in a range of cases, the cohomology is a free module over the cohomology of the structure sheaf and that the space of generators is an irreducible representation of the general linear supergroup that can be constructed via explicit multilinear operations. Our techniques come from commutative algebra: we relate this cohomology calculation to Tor groups of certain algebraic varieties.

💡 Research Summary

The paper “Borel‑Weil factorization for super Grassmannians” studies the sheaf cohomology of homogeneous vector bundles on the complex super Grassmannian
(X = \operatorname{Gr}(p|q,\mathbb C^{m|n})). In the classical (non‑super) setting every irreducible homogeneous bundle can be written as a tensor product of Schur functors applied to the universal quotient bundle (Q) and the dual of the universal subbundle (R). The Borel‑Weil theorem then tells us that such a bundle has no higher cohomology and that its space of global sections is an irreducible representation of (\mathrm{GL}(m)).

The author seeks a super‑analogue of this picture. For a pair of partitions (\alpha,\beta) satisfying the inequalities
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