`Tensor product decomposition for the general linear supergroup $GL(m|n)$'

Abstract
$\mathfrak{gl}(n)$ denote the Lie algebra of the general linear
group $GL(n)$. Given two finite dimensional irreducible
representations $L(\lambda), L(\mu)$ of $\mathfrak{gl}(n)$, its tensor
product decomposition $L(\lambda) \otimes L(\mu)$ is given by the
Littlewood-Richardson rule.

The situation becomes much more complicated when one replaces
$\mathfrak{gl}(n)$ by the general linear Lie superalgebra
$\mathfrak{gl}(m|n)$. The analogous decomposition $L(\lambda) \otimes
L(\mu)$ is largely unknown. Indeed many aspects of the representation
theory of $\mathfrak{gl}(m|n)$ are more akin to the study of Lie
algebras and their representations in prime characteristic or to the
BGG category $\mathcal{O}$. I will give a survey talk about this
problem and explain why some approaches don't work and what can be
done about it. This will give me the chance to speak about a) the
character formula for an irreducible representation $L(\lambda)$, b)
Deligne's interpolating category $Rep(GL_t)$ for $t \in \mathbb{C}$
and c) the process of semisimplification of a tensor category.