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

Abstract

We establish an explicit formula for the character of an irreducible finitedimensional representation of gl(m|n). The formula is a finite sum with integer coefficients in terms of a basis E µ (Euler characters) of the character ring. We prove a simple formula for the behaviour of the "superversion" of E µ 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).

On modified extension graphs of a fixed atypicality

“…We will present the corresponding Poincaré polynomials K λ,ν (z) := i K i (λ; ν)z i in terms of the arch diagram. The same Poincaré polynomials appear in the character formulae obtained in [22], [4], [35] and [16] (in particular, the arch diagrams in q m -case are similar to the diagrams appeared in [35], 3.3). The multiplicites…”

“…Remark. The coefficients of the character formulae obtained in [22], [35], [16] can be expressed in terms of the values K λ,ν (−1). By above, if K λ,ν (−1) = 0, then…”

The Duflo-Serganova functor, vingt ans après

“…Remark 12.6. There is an interesting link between arc diagrams and the computations of character formulas for gl(m|n) and osp(m|2n) (see [GH2]) as well as for q(n) (see [SuZh]). A similar connection is expected for p(n) as well.…”