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.



Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $W$ be a vector space of dimension $n$ containing $V$. Let $\alpha$ be an endomorphism of $V^{\otimes n}$ commuting with the action of ${\rm GL}(V)$. Suppose that $\alpha$ can be extended to an endomorphism $\beta$ of $W^{\otimes n}$ that commutes with the action of ${\rm GL}(W)$. Then, by the argument given by David Speyer in the question, there exist scalars $c\_\sigma \in \mathbf{C}$ such that


$$ \beta = \sum\_{\sigma \in S\_n} c\_\sigma \sigma $$


and this also expresses $\alpha$ as a linear combination of place permutations of the tensor factors. (As I noted in my comment, this expression is, in general, far from unique.)


Any proof that such an extension exists must use the
semisimplicity of $\mathbf{C}S\_n$, since otherwise we get an easy proof of general Schur-Weyl duality. If we assume that ${\rm GL}(W)$ acts as the full ring of 
$S\_n$-invariant endomorphisms of $W^{\otimes n}$ then a fairly short proof is possible. I think it is inevitable that it uses many of the same ideas as the double-centralizer theorem. A more direct proof would be very welcome.


Let $U$ be a simple $\mathbf{C}S\_n$-module appearing in $V^{\otimes n}$. Let


$$ X = U\_1 \oplus \cdots \oplus U\_a \oplus U\_{a+1} \oplus \cdots \oplus U\_b $$


be the largest submodule of $W^{\otimes n}$ that is a direct sum of simple $\mathbf{C}S\_n$-modules isomorphic to $U$. We may choose the decomposition so that $X \cap V^{\otimes n} = U\_1 \oplus \cdots \oplus U\_a$. Each projection map $W^{\otimes n} \rightarrow U\_i$ is $S\_n$-invariant, and so is induced by a suitable linear combination of elements of ${\rm GL}(W)$. 
Hence each $U\_i$ for $1 \le i \le a$ is $\alpha$-invariant. Similarly, for each pair $i$, $j$ there is a isomorphism $U\_i \cong U\_j$ induced by ${\rm GL}(W)$; these isomorphisms are unique up to scalars (by Schur's Lemma). Using these isomorphisms we get a unique ${\rm GL}(W)$-invariant extension of $\alpha$ to $X$.


Finally let $W^{\otimes n} = C \oplus D$ where $C$ is the sum of all simple $\mathbf{C}S\_n$-submodules of $W^{\otimes n}$ isomorphic to a submodule of $V^{\otimes n}$ and $D$ is a complementary $\mathbf{C}S\_n$-submodule. The previous paragraph extends $\alpha$ to a map $\beta$ defined on $C$. The projection map $W^{\otimes n} \rightarrow D$ is $S\_n$-invariant and so is induced by ${\rm GL}(W)$. Hence we can set $\beta(D) = 0$ and obtain a ${\rm GL}(W)$-invariant extension $\beta : W^{\otimes n} \rightarrow W^{\otimes n}$ of $\alpha$. 


