$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SW{SW}$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?

`$p$`

continues to attract creative interest. Following his suggestion, the MathSciNet reference to the paper by Carter-Lusztig is MR0354887 (50 #7364) 20G05 (20C20) Carter, Roger W.; Lusztig, George, On the modular representations of the general linear and symmetric groups. Math. Z. 136 (1974), 193–242. (One of two substantial papers they wrote during Lusztig's short time at Warwick.) . $\endgroup$