Frobenius series for the $S_n$-module $\mathbb{Q}[X]$

Question

I'm reposting this question, by recommendation of a moderator.

I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants. In what follows, all vector spaces and algebras are over the field of rational numbers $\mathbb{Q}$.

First, let $\phi$ be the Frobenius characteristic map which assigns to each character $\chi$ of $S_n$ a symmetric function $\phi(\chi)$ such that if $\chi_\lambda$ is the character of the irreducible representation of $S_n$ indexed by a partition $\lambda$ of $n$, then $\phi(\chi_\lambda)$ is the Schur function $s_\lambda$ indexed by the same partition $\lambda$. If $V$ is the representation of $S_n$ whose character is $\chi$, we abuse notation and write $\phi(V)$ instead of $\phi(\chi)$.

Let $A=\bigoplus_{i\geq 0} A_i$ be a graded $S_n$-module. We define the Frobenius series of $A$ by $$ F_A(q) = \sum_{i\geq 0} \phi(A_i) q^{i}. $$ Similarly, if $A=\bigoplus_{i,j\geq 0} A_{ij}$ is a bigraded $S_n$-bimodule, we define its Frobenius series by

$$ F_A(t,q) = \sum_{i,j\geq 0} \phi(A_{ij}) t^{i}q^{j}. $$

In Section 1.4 of the cited article, it is mentioned that for $\mathbb{Q}[X]=\mathbb{Q}[x_1,\dotsc,x_n]$ (with the natural action of $S_n$ permuting the variables), the corresponding Frobenius series is given by $$ F(q) = \sum_{\lvert\lambda\rvert = n} s_\lambda(1,q,q^2,\dotsc)s_\lambda(z_1,z_2,\dotsc) = \left.\prod_{i,j} \frac{1}{1-q^{i}z_ju}\right\rvert_{u^n}. $$ Here the notation $f(u)\rvert_{u^n}$ means to take the coefficient of $u^n$ in the series $f(u)$.

All that is said in the mentioned article, is that this is a well-known computation using MacMahon's master theorem. I'm new in this subject so this is not well-known to me at all, so if you can suggest me some reference where I can find this computation, it would be great!

Finally, in the same section, it is mentioned that for the bigraded ring $\mathbb{Q}[X,Y]=\mathbb{Q}[x_1,\dotsc,x_n,y_1,\dotsc,y_n]$, where $S_n$ acts by permuting the pairs of variables $(x_i,y_i)$, the corresponding Frobenius series is given by $$ F(t,q) = \sum_{\lvert\lambda\rvert=n}\sum_{\lvert\mu\rvert=n} s_\lambda(1,t,t^2,\dotsc)s_\mu(1,q,q^2,\dotsc)(s_\lambda\ast s_\mu)(z_1,z_2,\dotsc) = \left.\prod_{i,j,k} \frac{1}{1-t^{i}q^{j}z_k u}\right\rvert_{u^n} $$ where $s_\lambda\ast s_\mu = \phi(\chi_\lambda\otimes \chi_\mu)$. I don't know where to find this computation either.

ADDED: I know how to obtain the second equality in each formula (that is, expressing the series with symmetric function coefficients as coefficients of the corresponding infinite products). My question is about how to obtain the explicit expressions for the Frobenius series $F(q)$ and $F(t,q)$

Answer 1

I have some minor resources for this. For example, on representation theory, and some general info about diagonal harmonics.

Now, I think that a good source of information is Jim Haglunds book,

James Haglund. The -Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007

The first identity you have there, is due to the Cauchy identity for Schur functions. This identity is a consequence of the RSK bijection.

The last identity, involves Kronecker coefficients (in a certain sense).

I am no expert on how to get that the Frobenius image becomes those symmetric functions though, perhaps someone else can give a good reference, but I know there are a few pages of this in Jim's book.