$\newcommand{\Sym}{\mathfrak S_3}$ $\newcommand{\Z}{\mathbb Z}$ $\newcommand{\C}{\mathbb C}$
I am considering the topological group $G = \Sym^\Z$ (where $\Sym$ is the symmetric group on $3$ letters) with the topology induced by the Cantor distance ($d(x, y) = 2^{- \min\{|i| \in \Z \,|\, x_i \neq y_i\}}$), i.e. the prodiscrete topology. I am interested in complex, (continuous) unitary, irreducible representations of this group. Let $\phi : \Sym^\Z \to GL_n(\C)$ be such a representation.
The continuity assumption tells us that $\phi(x)$ only depends on a finite number points, that is, $\exists V \in \Z, V \text{ finite }, \forall x, y \in \Sym^\Z, (\forall i \in V, x_i = y_i) \Longrightarrow \phi(x) = \phi(y)$. For simplicity of the notations, I will now assume that $\phi : \Sym^V \to GL_n(\C)$.
For $x \in \Sym^V$, let us write $e_i(x) : j \mapsto \begin{cases}x_i &\text{ if } j=i\\ id_{\Sym} &\text{ otherwise } \end{cases}$. This is simply the sequence equals to $x_i$ in the $i-$th position, and $id_{\Sym}$ everywhere else.
Now, by writing each $x \in \Sym^V$ as the product over $V$ of the $e_i(x)$, and as $\phi$ is a morphism, we can see that $\phi(x) = \phi(\prod\limits_{i \in V} e_i(x)) = \prod\limits_{i \in V} \phi(e_i(x)) = \prod\limits_{i \in V} \phi_i(x_i)$ where the $\phi_i$ are defined in a straightforward manner. All the $\phi_i$ are representations of the symmetric group $\Sym$, although not necessarily irreducible.
An important remark is that as the product of the $e_i(x)$ is commutative (meaning that it can be computed in any order), so is the product of the $\phi_i(x_i)$. Hence, we do have that
$\forall i \neq j \in V, \forall \sigma, \tau \in \Sym, \phi_i(\sigma)\phi_j(\tau) = \phi_j(\tau)\phi_i(\sigma)$
This property seems very restrictive, as it is asbolutely not verified in general for 2 representations of $\Sym$, even if they are reducible. My question is: what can we say about this product/the $\phi_i$ ? For example, for $i \neq j$, and any $\sigma, \tau$, $\phi_i(\sigma)$ and $\phi_j(\tau)$ are co-diagonalizable. But what does this is actually imply ? Can we have several (an arbitrary large number ?) different $\phi_i$ whose decomposition in a direct sum of irreducible representations contain an occurrence of the standard representation of $\Sym$ ? Can we have an arbitrarily large number of $\phi_i$ who are not the identity of $GL_n(\C)$ ?
Any insight, or any reference to a book/article/whatever is very welcome !
Thank you !
P.-S.: an additional remark is that we can decompose $\phi$ as a tensor product of irreducible representations (up to isomorphism). As the irreducible rep. of $\Sym$ are of dimension 1 or 2 only, this means that the resulting dimension of the tensor product is of dimension $2^k$ for some $k$. If anyone has a other, more intuitive or "combinatoric" proof of this result, I am very willing to hear it, as I don't see why we could have irreducible representations of $\Sym^\Z$ of degree 8 or 16 but not 6 or 12 !
Edit: clarified a few notations, and typos.