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 g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) . The formula is a finite sum with integer coefficients in terms of a basis E μ \mathcal{E}_{\mu} (Euler characters) of the character ring. We prove a simple formula for the behavior of the “superversion” of E μ \mathcal{E}_{\mu} in the g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) -case under the map ds \mathrm{ds} on the supercharacter ring induced by the Duflo–Serganova cohomology functor DS \mathrm{DS} . As an application, we get combinatorial formulas for superdimensions, dimensions and g 0 \mathfrak{g}_{0} -decompositions for g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) .