Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, any permutation of the tensor factors is in $A$. I am looking for an elementary proof that these permutations span $A$.

If $\dim V \geq n$, there is a very simple proof. Take $e_1$, $e_2$, ..., $e_n$ in $V$ linearly independent and let $\alpha \in A$. Then $\alpha(e_1 \otimes e_2 \otimes \cdots \otimes e_n)$ must be a $t_1 t_2 \cdots t_n$ eigenvector for the action of the matrix $\mathrm{diag}(t_1, t_2, \ldots )$ in $GL(V)$. So $\alpha(e_1 \otimes \cdots \otimes e_n) = \sum_{\sigma \in S_n} c_{\sigma} e_{\sigma(1)} \otimes \cdots \otimes e_{\sigma(n)}$ for some constants $c_{\sigma}$. It is then straightforward to show that $\alpha$ is given by the corresponding linear combination of permutations.

I feel like there should be an elementary, if not very well motivated, extension of the above argument for the case where $\dim V < n$, but I'm not finding it.

**Motivation:** I'm planning a course on the combinatorial side of $GL_N$ representation theory -- symmetric polynomials, jdt, RSK and, if I can pull it off, some more modern things like honeycombs and crystals. Since it will be advertised as a combinatorics course, I want to prove a few key results that give the dictionary between combinatorics and representation theory, and then do all the rest on the combinatorial side. Based on the lectures I have outlined so far, I think this will be one of the few key results.

The standard proof is to show that the centralizer of $k[S_n]$ is spanned by $GL(V)$, and then apply the double centralizer theorem. Although the double centralizer theorem (at least, over $\mathbb{C}$) doesn't formally involve anything I won't be covering, I think it is pretty hard to present it to people who aren't extremely happy with the representation theory of semi-simple algebras. So I am looking for an alternate route.

functorsrather than Schur modules, i. e., considering $V$ as a variable rather than a fixed vector space. In that case you can more-or-less WLOG assume $\dim V\geq n$ and use your argument. Of course, the result you thus prove is weaker, but the question is whether your goal is the classical Schur-Weyl duality of $\mathrm{GL}V$ for fixed $V$, or something else where Schur-Weyl duality is just a lemma (in the latter case, chances ... – darij grinberg Mar 3 '12 at 1:44`$\langle h_{\lambda}, m_{\mu} \rangle = \delta_{\lambda \mu}$`

, this comes down to computing`$\mathrm{Hom}(\mathrm{Sym}^{\lambda_1} \otimes \cdots \otimes \mathrm{Sym}^{\lambda_n}, \mathrm{Sym}^{\mu_1} \otimes \cdots \otimes \mathrm{Sym}^{\mu_n})$`

. If you know the above claim, this turns into some very nice combinatorics, and leads into RSK in a clean way. – David Speyer Mar 3 '12 at 13:57