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

First, let $\phi$ the Frobenius characteristic map which assings 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 $\mathbb{Q}[X]=\mathbb{Q}[x_1,\dots,x_n]$ (with the natural action of $S_n$ permuting the variables), the corresponding Frobenius series is given by $$ F(q) = \sum_{|\lambda| = n} s_\lambda(1,q,q^2,\dots)s_\lambda(z_1,z_2,\dots) = \left.\prod_{i,j} \frac{1}{1-q^{i}z_ju}\right|_{u^n} $$ Here the notation $f(u)|_{u^n}$ 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 $\mathbb{Q}[X,Y]=\mathbb{Q}[x_1,\dots,x_n,y_1,\dots,y_n]$, where $S_n$ acts by permuting the pairs of variables $(x_i,y_i)$, the corresponding Frobenius series is given by $$ F(t,q) = \sum_{|\lambda|=n}\sum_{|\mu|=n} s_\lambda(1,t,t^2,\dots)s_\mu(1,q,q^2,\dots)(s_\lambda\ast s_\mu)(z_1,z_2,\dots) = \left.\prod_{i,j,k} \frac{1}{1-t^{i}q^{j}z_k u}\right|_{u^n} $$ where $s_\lambda\ast s_\mu = \phi(\chi_\lambda\otimes \chi_\mu)$. I don't know where to find this computation either.

Thank you in advance!



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