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$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SW{SW}$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?
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    $\begingroup$ The keyword is "Schur algebra" (see the work of Green, Dipper, James, Donkin and a book by Martin). $\endgroup$ Jun 3, 2010 at 1:05
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    $\begingroup$ To second Victor's comment, Schur (and then $q$-Schur) algebras grow out of Schur-Weyl duality in char $p$; this allows one in particular to use finite dimensional algebra techniques for the study of modular representations of general linear groups. Donkin's papers and books are an important source, but by now the literature is large. For instance there is an added chapter on "Truncated Categories and Schur Algebras" in the 2003 AMS second edition of Jantzen's Representations of Algebraic Groups. $\endgroup$ Jun 3, 2010 at 18:17
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    $\begingroup$ This (modular Schur-Weyl duality) is the topic of Carl Mautner's beautiful 2010 PhD thesis. He obtains the Schur functor geometrically from a geometric relation between the affine Grassmannian for GL_n (where reps of GL_n sit, by the Geometric Satake theorem) and the nilpotent cone for gl_n (where reps of S_n sit, by Springer theory), recovering in particular the modular Schur-Weyl duality theorem of Carter and Lusztig (which ought to be mentioned here independently). $\endgroup$ Jun 3, 2010 at 23:55
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    $\begingroup$ As David indicates, Schur-Weyl duality in 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.) . $\endgroup$ Jun 4, 2010 at 15:48

2 Answers 2

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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.

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  • $\begingroup$ Before I posted my questions, I have looked at the paper by Stephen Doty roughly. It seems it doesn't give the answer for what I want immediately. $\endgroup$
    – JJH
    Jun 2, 2010 at 21:56
  • $\begingroup$ See my edited answer for more details. But the subject is spread out, so "immediate" answers may be scarce at this point. $\endgroup$ Jun 2, 2010 at 22:44
  • $\begingroup$ It seems we also have functor from modular representation of $S_k$ to modular representation of finite group of Lie type(representation of $G(k)$ over $\bar{k}$ , where $k$ is a finite field), but isn't it more natural to use representation of Frobenius kernel or hyperalgebra? There are supposed to be some relations with these two different type of representations? $\endgroup$
    – JJH
    Jun 3, 2010 at 5:25
  • $\begingroup$ I tried to give a short answer to the summary question: For my purpose, what is the correct formulation for schur-Weyl duality in positive characteristic? But to get into the current state of modular representations of general linear or other reductive groups requires the machinery developed in Jantzen's book mentioned above along with related papers. Restricting to finite subgroups or to Frobenius kernels is a major theme but goes far beyond the study of Schur-Weyl duality in char $p$. (For finite groups, there is also a lot of work on other primes dividing the group order.) $\endgroup$ Jun 3, 2010 at 18:23
  • $\begingroup$ @Jim, May I ask a question here? Over positive characteristic, Given reductive group $G$, is $\mathcal{O}(G)$ still isomorphic to $\oplus V_\lambda\otimes V_\lambda ^*$ as $G\times G$ representatioin? Here $V_\lambda $ is Weyl module or dual Weyl module. Another question is, On space $\otimes ^k F^n$ , we have action of $GL_n\times S_k$ , if $k<n$ , can we alway decompose $\otimes^k F^n$ as $\oplus V_\lambda \otimes \sigma_\lambda $, where $V_\lambda $ is Weyl module, and $\sigma_\lambda$ is the Specht module. Thank you very much. $\endgroup$
    – JJH
    Jun 17, 2010 at 18:39
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(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.

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