Gruson-Serganova character formulas and the Duflo-Serganova cohomology functor

Abstract

We establish an explicit formula for the character of an irreducible finite-dimensional representation of $\mathfrak{gl}(m|n)$. The formula is a finite sum with integer coefficients in terms of a basis ${\mathcal{E}}_{\mu }$ (Euler characters) of the character ring. We prove a simple formula for the behaviour of the ``superversion'' of ${\mathcal{E}}_{\mu }$ in the $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$-case under the map $ds$ on the supercharacter ring induced by the Duflo-Serganova cohomology functor $DS$. As an application we get combinatorial formulas for superdimensions, dimensions and ${\mathfrak{g}}_0$-decompositions for $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$.