I've been looking into the Quantum Schur Transform, first introduced by this paper.
The explanation of the procedure goes like so:
Consider the two groups: $S(N)$, the symmetric group on $N$ elements, and $U(D)$, the unitary group in $D$ dimensions.
We have the two representations:
$$ P(s) | i_1, i_2, \cdots, i_N \rangle = | i_{s^{-1} (1)}, i_{s^{-1} (2)}, \cdots, i_{s^{-1} (N)} \rangle $$
where $s \in S(N)$ is an arbitrary permutation of $N$ elements
$$ Q(u) | i_1, i_2, \cdots, i_D \rangle = u | i_1 \rangle \otimes u | i_2 \rangle \otimes \cdots \otimes u | i_D \rangle $$
where $u \in U(D)$.
In both of these representations, each $| i_x \rangle$ is a $d$ dimensional vector (the same dimension of the unitary group).
Both representations are reducible, and can be written as:
$$ P(s) \cong \bigoplus_\alpha I_{n_\alpha} \otimes p_\alpha(s) $$
$$ Q(s) \cong \bigoplus_\beta I_{m_\beta} \otimes q_\beta(s) $$
Where $\alpha$ and $\beta$ are labels for irreps of $S(N)$ and $U(D)$ respectively, and $\cong$ is equality up to a unitary change of basis.
I don't understand what the $I$, $n_\alpha$ and $m_\beta$ terms are here. The two equations are apparently an isotypic decomposition, and $n_\alpha$ and $m_\beta$ are known as "multiplicities" (according to Eq. 1), but I can't really find any resource on the topic of isotypic decomposition and multiplicity that take the same form as the equations above. And I don't really know what the $I$ term is meant to represent.
I am trying to give an example to illustrate the situation in the human perspective that helped me to have a picture of two commuting representations on the same space. In representation theory, there is a group $G$ acting on the space $V$. Usually $V$ is tacitly understood and objects related to $V$ are denoted to tacitly match the obvious. Here, the obvious is the problem, so i will tacitly reduce the $G$-information to a maximum, but invest a lot of details on the $V$-part. Let $C$ be the vector space $\Bbb C$, i need a notation to be quickly typed. Now consider the spaces $V=C^7$ and $W=C^3$. We need a picture for $V$, for $W$, and for $V\otimes W$, seen as vector spaces over $C$ with no other structure. (No group action.) Here is the picture: $$ V= \begin{bmatrix} C & C & C & C & C & C & C \end{bmatrix}\ ,\qquad W=\begin{bmatrix} C \\ C \\ C \end{bmatrix}\ ,\\ V\otimes W = \begin{bmatrix} C & C & C & C & C & C & C\\ C & C & C & C & C & C & C\\ C & C & C & C & C & C & C \end{bmatrix}\ . $$ Now we are enriching the structure letting the symmetric group $S(N)$ act on $V$, we permute the components. Denote the representation by $\pi$. Here $N$ is seven, but the picture is easily extrapolated in the brain. (Depending on the choice of the side, left action or right action, we may need to let a permutation act by itself or its inverse.) And on $W$ we have the corresponding $U(D)$ acting, here $D$ is three. Denote the corresponding representation by $\rho$. Do we have an action of the product group on "something related", yes, we have, look at the following picture: $$ \text{Action of $U(D)$ on columns } \Bigg\{\ \underbrace{ \begin{bmatrix} C & C & C & C & C & C & C\\ C & C & C & C & C & C & C\\ C & C & C & C & C & C & C \end{bmatrix}}_{\displaystyle\text{Action of $S(N)$ on rows}} \ \ . $$ Explicitly, as a part of notation, we denote by $\pi\otimes 1:V\otimes W\to V\otimes W$ the new representation of $S(N)$ on $V\times W$, it is induced by $\pi$ by tensor product with the identity. Well, this one is in representation theory not really wanted, since there is a trivial representation, that may be also denoted by one. So people insist to use something else. E.g. $I$ for the identity matrix, or a
\mahbb 1that is looking like $\Bbb I$, but is an one symbol.The same for the representation $\rho$, it induces $1\otimes\rho$ on $V\otimes W$.
To have even more decorations to distract the attention, people insist to give the dimension of the space where the identity $I$ is acting on, so we may see in our case $\pi\otimes I_D$ and $I_N\otimes \rho$.
Now my sample choice of $\pi$ is not an irreducible representation. The irreducible representations of $S(N)$ are known, they are parametrized by shapes of standard tableaux. There are many good books showing the combinatorics and the representation theoretical parts of the medal. OK, in our case we can split in the category of $S(N)$-representations into irreducible pieces: $$ \pi=\bigoplus_{\alpha\in A}\pi_\alpha\ , $$ where we may think that $\alpha$ runs in the special list $A$ of shapes of tableaux of total size $N$. But in general we may not (want to) have an explicit description, so we consider tacitly that we know how to parametrize the irreducible representations, and the index set for this is often omitted. Similarly, in the category of $U(D)$-representations we may want to split into irreducible pieces: $$ \rho=\bigoplus_{\beta\in B}\rho_\beta\ , $$ and there is no closer description of $B$.
Now the splitting of $\pi:V\to V$ induces a splitting of $\pi\otimes I$ on $V\otimes W$, functorially we have: $$ \pi\otimes I=\bigoplus_{\alpha\in A}\pi_\alpha\otimes I\ , $$ and one may want to add the decoration to $I$ to also collect the dimension of $W$, which is $D$, so write $I_D$ instead of $I$ above, if wanted. Similarly, one has a splitting $$ I\otimes \rho=\bigoplus_{\beta\in B}I\otimes\rho_\beta\ , $$ and one may want or not to have some index $N$ for $I$, to show it corresponds to some $N\times N$ unit matrix, or the identity (linear map) of a space of dimension $N$.
Let us now get the bridge to the question. I will still use $\pi,\rho$ instead of the notations $P,Q$ from the question.
In our case, the two representations commute, so we may look at the representation $\pi\otimes \rho$ of $S(N)\times U(D)$ acting on $V\otimes W$, and there is a "package" split $$ \pi\otimes \rho=\bigoplus_{\substack{\alpha\in A\\\beta\in B}} \pi_\alpha\otimes\rho_\beta\ . $$ Now we come to the initial situation of the question. We do not have the above picture of a tensor product space $V\otimes W$, (but we want it,) instead we have one space, say $\Bbb V$, with representations, and want to bring it into this "tensor shape" if possible. So we split $\Bbb V$ in irreducible $\alpha$-components. However, each component may have a different dimension involved, so this is the reason for writing the lower index decoration $n_\alpha$ at $I$ for the component $\pi_\alpha$. And since we have no "tensor pieces", and no order for them, the splitting is existential, and up to conjugation we may place the $I$-part on the first place, thus writing $\pi:\Bbb V\to\Bbb V$ as $\bigoplus I_{n_\alpha}\otimes\pi_\alpha$.
The same applies for $\rho$. We obtain then a splitting of $\Bbb V$ as $$ \bigoplus_{\substack{\alpha\in A\\\beta\in B}} I_{m_{\alpha\beta}}\otimes\pi_\alpha\otimes\rho_\beta\ , $$ relations $(7)$, $(8)$ in loc. cit. - and the paper comes to the conclusion that under the circumstances specified there, the multiplicities $(m_{\alpha\beta})$ are "simple", the $(\alpha,\beta)$-component $\pi_\alpha\otimes\rho_\beta$ is either missed, or it appears with multiplicity one.