Cohomological Tensor Functors on Representations of the General Linear Supergroup

Abstract

We define and study cohomological tensor functors from the category $T_n$ of finite-dimensional representations of the supergroup $Gl(n|n)$ into $T_{n-r}$ for $0<r\le n$. In the case $DS:T_n\to T_{n-1}$ we prove a formula $DS(L)=⨁{\Pi }^{n_i}{L}_i$ for the image of an arbitrary irreducible representation. In particular $DS(L)$ is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation.

arXiv commentv5: Added two sections on the cohomology of Kac-Modules. To appear in Memoirs of the American Mathematical Society v4: Extended results on the cohomology and the forest formula. Corrected typos and grading conventions. Section numbering has changed. v3:Corrected typos. Changed a sign normalization in section 14