Tensor Spaces and Commuting Actions
We want to build up to a discussion of Schur-Weyl duality, so for the moment we will depart from our discussion of the representation theory of the symmetric group and discuss a space on which both the symmetric and general linear groups act: the tensor product space.
Consider a complex vector space $V$ of dimension $d$, whose basis we take to be $\lbrace e _ {1}, \cdots, e _ {d}\rbrace$. From $V$, we construct the $n$-fold tensor product space, denoted $V^{\otimes n}$, whose typical basis element looks like $ e _ {i _ {1} } \otimes \cdots \otimes e _ {i _ {n} } $, where each index $i _ {k}$ can range from $1$ to $d$. The dimension of this space is therefore $d ^ {n} $.
As a concrete example, let $V$ be a $2$-dimensional space with basis ${e _ {1}, e _ {2}}$. If we consider the $2$-fold tensor product $V^{\otimes 2}$, it has dimension $2 ^ {2} = 4$. Its basis consists of: \begin{equation} \left\lbrace e _ {1} \otimes e _ {1} , e _ {1} \otimes e _ {2} , e _ {2} \otimes e _ {1} , e _ {2} \otimes e _ {2} \right\rbrace \ . \end{equation} Any general tensor in this space is just a linear combination of these four basis vectors.
Right Action of Symmetric Group
The symmetric group $S _ {n}$ naturally acts on $V ^ {\otimes n}$ by permuting the indices. We define this to be a right action — this will be important to keep in mind. If we have a simple tensor $v _ {1} \otimes \cdots \otimes v _ {n}$ and a permutation $\sigma \in S _ {n}$, the group acts from the right by permuting the indices of the vectors: \begin{equation} (v _ {1} \otimes \cdots \otimes v _ {n}) \cdot \sigma = v _ {\sigma(1)} \otimes \cdots \otimes v _ {\sigma(n)} \ . \end{equation}
Let us see this in action. Consider $V ^ {\otimes 3}$ and the permutation $\sigma = (123) \in S _ {3}$. If we apply this to a basis tensor $e _ {1} \otimes e _ {2} \otimes e _ {1}$, we get \begin{equation} (e _ {1} \otimes e _ {2} \otimes e _ {1}) \cdot (123) = e _ {1} \otimes e _ {1} \otimes e _ {2} \ . \end{equation} Notice that the permutation moves the vectors into new slots, completely independently of what those vectors actually are.
Left Action of General Linear Group
Now let us bring in the general linear group, $GL(V)$, which consists of all invertible linear transformations (or invertible $d \times d$ matrices) acting on $V$.
While $S _ {n}$ moves the vectors in a tensor product around, $GL(V)$ transforms the vectors themselves. We define a left diagonal action of $GL(V)$ on $V^{\otimes n}$ as follows: for any matrix $g \in GL(V)$, its action on a simple tensor is defined by acting on every single tensor factor simultaneously as \begin{equation} g \cdot (v _ {1} \otimes \cdots \otimes v _ {n}) = (gv _ {1}) \otimes \cdots \otimes (gv _ {n}) \ . \end{equation} To illustrate, let us return to our $d=2$ example in $V^{\otimes 2}$. Suppose we apply a generic $2 \times 2$ matrix
\[g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\]to the basis tensor $e _ {1} \otimes e _ {2}$. First, note how $g$ acts on the individual basis vectors: $ge _ {1} = a e _ {1} + c e _ {2}$ and $g e _ {2} = b e _ {1} + d e _ {2}$. Now apply it to the tensor:
\[\begin{align} g \cdot (e _ {1} \otimes e _ {2}) &= (ge _ {1}) \otimes (ge _ {2}) = (a e _ {1} + c e _ {2}) \otimes (b e _ {1} + d e _ {2}) \ , \\ &= a b (e _ {1} \otimes e _ {1}) + a d (e _ {1} \otimes e _ {2}) + c b (e _ {2} \otimes e _ {1}) + c d (e _ {2} \otimes e _ {2}) \ . \end{align}\]Commuting Actions
We now have two distinct group actions on the tensor product space $V^{\otimes n}$: $GL(V)$ acting from the left by transforming the vectors themselves, and $S _ {n}$ acting from the right by shuffling the vectors themselves.
It should be evident at this point that these two actions commute; it does not matter whether one first transforms the vectors and then shuffle them, or one shuffles their positions first and then transforms them. Said differently, for any $g \in GL(V)$, any $\sigma \in S_n$, and any tensor $v \in V^{\otimes n}$: \begin{equation} g \cdot (v \cdot \sigma) = (g \cdot v) \cdot \sigma \ . \end{equation} The proof is straightforward and is left as an exercise.
The commuting actions here lie at the heart of Schur-Weyl duality. Because the algebra generated by $S _ {n}$ and the algebra generated by $GL(V)$ commute within the space of linear maps on $V^{\otimes n}$, they act as what are called “centralisers” for one another. In turn, this links irreducible representations of both groups.