Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$.
For every natural number $d$ we have a natural map $$\Phi\_d:\mathbb{C} S\_d\to \text{End}\_G(V^{\otimes d})$$
where $\sigma\in S\_d$ is sent to the linear transformation given by tensor permuting $V^{\otimes d}$ according to $\sigma$. To what extent does Schur-Weyl duality generalise from the case $n=0$ to the case of super vector space? That is:


**Question 1:** Is $\Phi\_d$ surjective?


**Question 2:** What is the kernel of $\Phi\_d$ ? Can it also be described using some combinatorial condition similar to the case $n=0$?



Schur Weyl duality for the supergroup $\text{GL}(m|n)$
Schur Weyl duality holds in the super case, as well. There is the double centralizer property, thus a positive answer to Q1, and also a characterization of the kernel as those ideals of $\mathbb C[S\_d]$ which correspond to partitions that don't fit inside the (m,n)-hook.


See the paper ["Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras"](https://core.ac.uk/download/pdf/82438431.pdf) by Berele and Regev. For a more recent textbook treatment this is also done nicely in chapter 11 of Musson's book "Lie Superalgebras and Enveloping Algebras".


