Schur-Weyl duality in positive characteristic $\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SW{SW}$Let $S_k$ be the symmetric group. Let $F$ be an algebraically closed field. Let $\Rep(S_k)$ be the category of representations of $S_k$ over $F$.
Let $\Rep(\GL_n(F))$ be the category of algebraic representations of $\GL_n(F)$.
We can construct a functor $\SW$ from $\Rep(S_k)$ to $\Rep(\GL_n(F))$, $\SW(\sigma)=(\otimes^k F^n\otimes \sigma)^{S_k}$, where $\sigma\in \Rep(S_k)  $.
When $F$ is of characteristic 0, and n>k, it is well-known $\SW$ is a fully faithful and exact functor. Usually it is called Schur-Weyl duality.
My question is:

*

*When $F$ is of characteristic $p$, is
$\SW$ still a fully faithful and exact
functor?

When $p>k$, the representation theory of $S_k$ over $F$ behaves exactly the same as characteristic 0 case, but for algebraic representation of $\GL_n(F)$, it is totally different from characteristic 0 case.
When $p<=k$, representation of $S_k$ is complicated.
It is well-known problem, to determine the decomposition number in $\Rep(S_k)$.

*

*Is it possible to use Schur-Weyl
duality to determine the decomposition number for $S_k$?

Since for modular representation theory of reductive group, the similar problem is known or almost known by Kazhdan-Lusztig polynomial.
TO sum up, I would like to ask:

*

*For my purpose, what is the correct formulation for
Schur-Weyl duality in positive
characteristic?

 A: The questions here have certainly been explored (though not definitively) in many recent papers or preprints on arXiv.    Look for example at the arXiv paper by Stephen Doty Link, as well as many others by Steve and/or his collaborators.   Most of the arXiv papers have subject listing RT (some also consider quantum analogues under QA).  But some predate arXiv; there has been a lot of study of decomposition numbers of symmetric groups in prime characteristic, for example, using what little is known about modular representations of GL$_n$.   Not having gone far with this literature myself, I'd suggest that you start the inquiry with available papers and then maybe raise narrower questions here.
My main point at first has been that a lot of literature exists from the past couple of decades, so the questions should focus on what is in that literature (not just Doty's short conference paper I cited).   Concerning the status of modular representations for finite groups of Lie type, what's known is not yet good enough to answer most questions about symmetric groups for small primes.   While Lusztig's conjectures promise a good conceptual picture for primes at least the Coxeter number of the Weyl group, even that much can be implemented only in recursive style.   For small primes little is known, but it would have immediate applications to $S_n$. In classical Schur-Weyl duality the dictionary goes the opposite way.
A: (This seems to be an old question, but just recently edited, so it came up today.)
The condition that $n \geq k$ means that the polynomial representations of degree $k$ in the algebraic representations of $GL_n(\mathbb F)$ agree with the category $\mathcal P^k(\mathbb F)$ of `strict polynomial functors of degree $k$' as described by Suslin and Friedlander in the mid 1990's.  It is useful to view these categories, for all $k$, as a single object, and one can also very usefully work over any field (e.g. finite fields), not just algebraically closed ones.
The functor you wrote down is right adjoint to an exact functor $e_k: \mathcal P^k(\mathbb F) \rightarrow Rep_{\mathbb F}(S_k)$.  $e_k$ also has a left adjoint, and one is in the `recollement' setting.
In the finite field setting, I filter in the `kernel' of $e_k$ in my study of filtrations of these sorts of functor categories in [A stratification of generic representation theory and generalized Schur algebras, K-theory 26 (2002), 15-48].  This paper will also give you many references to relevent functor category literature.
