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

TL;DR

This paper provides explicit character formulas for irreducible representations of Lie superalgebras and explores their behavior under the Duflo-Serganova cohomology functor, leading to combinatorial formulas for superdimensions and decompositions.

Contribution

It introduces a new explicit character formula for $rak{gl}(m|n)$ and analyzes the supercharacter ring behavior under the $DS$ functor, with applications to superdimension and decomposition calculations.

Findings

Explicit finite sum formula for irreducible characters.

Simple behavior formula of supercharacters under $DS$ functor.

Derived combinatorial formulas for superdimensions and decompositions.

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 $\mathfrak{osp}(m|2n)$.