Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\dots,n_{t-2})$ be a $(t-1)$-tuple of integers and set $n_{t-1}:=-\sum\limits_{i=0}^{t-2}n_i$. Define $r_i=\sum\limits_{k=0}^{t-1}(n_k^2+\epsilon(i,k)n_k)/2$ for all $t$-residues $i$, where $\epsilon(i,k)=\left\{\begin{array}{rr}1,&\,\,\mbox{if $i\geq k$}\\-1,&\,\,\mbox{if $i<k$}\end{array}\right.$. Then there is a unique partition which is a $t$-core and which has $r_i$ nodes of $t$-residue $i$ [MR1055707 Reviewed Garvan, Frank; Kim, Dongsu; Stanton, Dennis Cranks and $t$-cores. Invent. Math. 101 (1990), no. 1, 1–17]. Notice that $n_i=r_i-r_{i+1}$ for each residue $i$.

Let $r(n):=(r_0,r_1,\dots,r_{t-1})\in{\mathbb Z}^t$ depend on $n:=(n_0,n_1,\dots,n_{t-2})$ as above and set $x^{r(n)}:=x_0^{r_0}x_1^{r_1}\dots x_{t-1}^{r_{t-1}}$ for indeterminates $x_0,\dots,x_{t-1}$. So the generating function for the $t$-residues of $t$-cores is $$ K_t(x):=\sum_{n\in{\mathbb Z}^{t-1}}x^{r(n)}. $$ Let $P(y)=\prod_n(1-y^n)^{-1}$ be the partition generating function in the indeterminate $y$ and let $J=(1,\dots,1)$ be the all-$1$'s vector in ${\mathbb Z}^t$. Using the representation of partitions on an abacus with $t$ runners, the generating function for the $t$-residues of all partitions is $$ P_t(x)=P(x^J)^tK_t(x). $$ According to Ben Elias, this is the generating function of the Fock space representation of $\hat{sl_t}$. So it should have a product expansion using Heisenberg and $\hat{sl_t}$ representation theory. For example, in a previous question [Generating function related to 2-residues of partitions I noted that the generating function for the $2$-residues of partitions factorizes as $$ \begin{aligned} P_2(x) &=P(x_0x_1)^2\sum_{n\in{\mathbb Z}}x_0^{n^2}x_1^{n^2+n}\\ &=\prod\limits_{n=1}^\infty\frac{(1+x_0^{2n-1}x_1^{2n})(1+x_0^{2n-1}x_1^{2n-2})(1+x_0^nx_1^n)}{(1-x_0^nx_1^n)}. \end{aligned} $$ This is a consequence of the Jacobi Triple Product Identity.

Question: can anyone explicitly factorize $P_t(x)$ and provide a self-contained explanation of the relevant representation theory?