Schur Weyl duality for the supergroup $\text{GL}(m|n)$

Question

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 $n=0$?

Answer 1

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" 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".

Answer 2

This is a result of Sergeev, but I can only find the article in Russian at the moment. If I remember right, the map is surjective, and the kernel can be worked out from the fact that a Schur functor applied to the standar representation of $\mathfrak{gl}(m|n)$ vanishes iff the Young diagram contains the box at position (m+1,n+1), i.e. if they do not fit in an "(m,n) hook." For example, see the discussion in these slides of Serganova. Note that the classical case the diagrams fitting in an (m,0)-hook are exactly the ones with m or fewer rows, while in the entirely odd case they need to have n or fewer columns.