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

We establish an explicit formula for the character of an irreducible finite-dimensional representation of gl(m|n). The formula is a finite sum with integer coefficients in terms of a basis epsilon(mu) (Euler characters) of the character ring. We prove a simple formula for the behavior of the "superversion" of epsilon(mu) in the gl(m|n)and 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 g(0)-decompositions for gl(m|n)and osp(m|2n).