TITLE: Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
QUESTION [23 upvotes]: 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.

REPLY [5 votes]: I'm going to write up Mark Wildon's proof as I understand it. As in the standard proof, we start by showing the Key Lemma that the centralizer of $k[S_n]$ is linearly spanned by $GL(V)$. Decompose $V^{\otimes n}$ into $S_n$-irreps, and let $\alpha$ be an endomorphism of $V^{\otimes n}$ commuting with $GL(V)$. For each irrep $U$ of $S_n$, let $U_1$, ..., $U_a$ be the occurrences of $U$ in $V^{\otimes n}$.
For any $U_i$, consider the endomorphism of $V^{\otimes n}$ which acts by $1$ on $U_i$ and on $0$ on all of the other summands of $V^{\otimes n}$. This commutes with $k[S_n]$ so, by the Key Lemma it is a linear combination of maps in $GL(V)$. Hence $\alpha$ commutes with it, which means that $\alpha$ takes $U_i$ to $U_i$ by some map $\alpha_i$. 
Consider the endomorphism of $V^{\otimes n}$ which takes $U_i$ to $U_j$ by an $S_n$-equivariant endomorphism and acts by $0$ on every other summand of $V^{\otimes n}$. This commutes with $k[S_n]$ so, by the Key Lemma it is a linear combination of maps in $GL(V)$. Hence $\alpha$ commutes with it, which means that $\alpha_i = \alpha_j$. (Abusing equals to mean "is taken to the other along the isomorphism $U_i \to U_j$, which is unique up to scalar".) Write $\alpha(U)$ for the common value of $\alpha_1$, $\alpha_2$, ..., $\alpha_a$.
There are now two ways to finish the proof.
Standard Argument: By Maschke and Artin-Wedderburn, there is an element in $k[S_n]$ which acts on each irrep $U$ by $\alpha(U)$. This element of $k[S_n]$ induces $\alpha$.
Mark Wildon's Argument: Let $V \subset W$. We will show that we can extended $\alpha$ to an endomorphism $\beta$ of $W^{\otimes n}$ which commutes with $GL(W)$. Decompose $W^{\otimes n}$ into $S_n$ irreps, so that the previous decomposition of $V^{\otimes n}$ occurs as a subset of the summands. Let the occurrences of $U$ be $U_1 \oplus U_2 \cdots \oplus U_a \oplus \cdots \oplus U_b$. Define a linear map $\beta:W^{\otimes n}\to W^{\otimes n}$ to act on all of the $U_i$ by $\alpha(U)$ or, if $a=0$ so that $\alpha(U)$ is undefined, define $\beta$ to act on the $U_i$ by $0$.
We claim that $\beta$ commutes with $GL(W)$. Proof: Any element of $GL(W)$ commutes with $k[S_n]$. So (by Schur's lemma), it can only map $U_i$ to a linear combination of other $U_j$'s, and the component of $\alpha$ mapping $U_i$ to $U_j$ is a scalar multiple of the standard isomorphism. Clearly, $\beta$ commutes with any map of this form.
Now, by my argument in the original post, take $\dim W \geq n$ to see that $\beta$ is induced by an element of $S_n$. Then $\alpha$ is also induced by this element of $S_n$.