Fix a finite set $X$ and two natural numbers $d$ and $n$.


For a partition $\lambda$ and a number $d$ denote by $s\_\lambda^d(x\_1,\dots,x\_d)$ the Schur polynomial in $d$-many variables $x\_1,\dots,x\_d$. Denote by $\mathcal P\_n(X)$ the set of partition-valued functions on $X$ of total size $n$, i.e. the set of functions $\lambda\colon X\to\{\text{partitions}\}$ such that $\|\lambda\|:=\sum\_{x\in X}|\lambda(x)|=n$.


Let $a\colon X\to \mathbb{N}\backslash\{0\}$ be a function on $X$. EDIT: I am mostly interested in the case where $a$ is the constant function with value $1$.


I am trying to evaluate the expression
$$F\_X(d,a):=\sum\_{\lambda\in \mathcal P\_n(X)}\left(\prod\_{x\in X}s^d\_{\lambda(x)}(a(x),\dots,a(x))\right)^2. $$ 




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EDIT: 


1. I have now good reason to believe that if we take for $a$ the constant function with value $1$ then we have $$F\_X(d,a)=\left(\binom{d^2|X|}{n}\right)$$
Here $\left(\binom{m}{n}\right)$ denotes the multiset number
2. More generally I am hoping that if $G$ is a finite group, $X$ is the set of irreducible characters (over $\mathbb C$) of $G$ and $a$ maps each character to its dimension then we have $$F\_X(d,a)=\left(\binom{d^2|G^{\mathrm{ab}}|}{n}\right)$$ Here $G^\mathrm{ab}$ is the abelianisation of $G$. This reduces to case 1 if $G$ is taken to be an abelian group.


EDIT2: my conjecture 2 above is totally wrong (as can be seen by hand in small examples like $G=S\_3$) and I have no good substitute hypothesis. I am now mostly interested in a proof of point 1 (where $a$ is the constant function $1$)




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The special case where $X=\{\star\}$ is a one-element set can be done using the identity
$$\prod\_{i,j}(1-x\_iy\_j)^{-1}=\sum\_n\sum\_{|\lambda|=n} s\_\lambda(x\_1,\dots)s\_\lambda(y\_1,\dots)$$
by putting $x\_i=y\_j=t$ for $i,j=1,\dots,d$ (and zero otherwise) and picking out the coefficient of $t^{2n}$ obtaining
$$c^d\_nt^{2n}=\sum\_{|\lambda|=n}s^d\_\lambda(t,\dots,t)^2,$$
where $c^d\_n$ is the coefficient of $t^n$ in $(1-t)^{-d^2}$ and can be calculated to be the multiset number $\left(\binom{d^2}{n}\right)$.
All in all we obtain $F\_{\{\star\}}(d,a)=\left(\binom{d^2}{n}\right)\cdot a(\star)^{2n}$ 



sum of squares of Schur polynomials indexed over partition valued functions on a set
assume $a$ is the constant function with value $1$.


as in the case of $X=\{\star\}$ we have the equality
$$(1-t^2)^{-d^2|X|}=\left(\sum\_{\lambda\in \{\mathrm{partitions}\}}s^d\_\lambda(t,\dots,t)^2\right)^{|X|}=\sum\_{\lambda\colon X\to\{\mathrm{partitions}\}}\prod\_{x\in X}s\_{\lambda(x)}^d(t,\dots,t)^2$$
and picking out the monomial $t^2$ gives
$$\left(\binom{d^2|X|}{n}\right)t^{2n}=\sum\_{\lambda\in P\_n(X)}\prod\_{x\in X}s\_{\lambda(x)}^d(t,\dots,t)^2$$
The result follows.


