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 qustion 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:
- What
- For my purpose, what is the correct formulation for schur Weyl duality in positive characteristic?
