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 $\mathfrak{ osp}(m|2n)$.
Categories: math.RT