Question

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:

• For my purpose, what is the correct formulation for schur Weyl duality in positive characteristic?

Answer 1

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 http://front.math.ucdavis.edu/0610.5591, 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.