TITLE: Arithmetic product of symmetric functions: why is it integral?
QUESTION [9 upvotes]: For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$.
We can define a bilinear map $\boxdot : \mathbf{Symm}_{\mathbb{Q}} \times \mathbf{Symm}_{\mathbb{Q}} \to \mathbf{Symm}_{\mathbb{Q}}$ by setting
$p_{\lambda} \boxdot p_{\mu} = \prod\limits_{i\geq 1,\ j\geq 1} p_{\operatorname*{lcm}\left(\lambda_i,\mu_j\right)}^{\gcd\left(\lambda_i,\mu_j\right)}$
for any two partitions $\lambda = \left(\lambda_1,\lambda_2,\lambda_3,...\right)$ and $\mu = \left(\mu_1,\mu_2,\mu_3,...\right)$. Here, we are writing $\boxdot$ as an infix operator (that is, $a\boxdot b$ means $\boxdot\left(a,b\right)$), and $p_\nu$ means the $\nu$-power sum symmetric function.
This bilinear map $\boxdot$ is associative. I call it the "arithmetic product", as it boils down to the arithmetic product of species viewed through the cycle index series.
Now, species theory can be used to show that $\boxdot$ restricts to a well-defined map $ \mathbf{Symm} \times \mathbf{Symm} \to \mathbf{Symm}$ (that is, the restriction of $\boxdot$ to $\mathbf{Symm} \times \mathbf{Symm}$ has its image in $\mathbf{Symm}$). My question is: Can this be proven more elementarily? Is there a good way to describe this map on an actual basis of $ \mathbf{Symm}$ rather than on the power-sum symmetric functions? Is there a more direct combinatorial or even representation-theoretical significance of this map?
(This is somewhat similar to MO question #120924, where another operation on $\mathbf{Symm}$ is constructed on the power sums first and then happens to be integral for weird reasons.)

REPLY [8 votes]: About the question
 "Is there a more direct combinatorial or even representation-theoretical significance of this map?"
Probably you know that the arithmetic product is an operation between representations of the symmetric group, because the species is a theory of representations of the symmetric group that gives concrete set theoretical constructions of operations, without the use of induced representations. The ordinary species correspond to permutation-representations and the tensor species to vector representations (representations on the general linear group). 
In the classical language, the arithmetic product of two represntations $\rho$ and $\tau$ of $S_m$ and $S_n$ respectively is given as the induced representation:
$$\rho\boxdot\tau=\mathrm{Ind}_{S_m\times S_n}^{S_{m.n}}\rho\otimes\tau.$$
But this seems to me less concrete than the direct recipe, given in the language of species, of the vector space where the symmetric group $S_U$ acts naturally,
$$(R\boxdot T)[U]=\bigoplus_{(\pi,\sigma)}R[\pi]\otimes T[\sigma]$$
The arithmetic product of two homogeneous symmetric functions in terms of the monomial s.f. is as follows
$h_m\boxdot h_n=\sum M_{\lambda} m_{\lambda}$
where $M_{\lambda}$ is the number of $m\times n$ matrices with $\lambda_i$ i-es (up to row and column permutations). This is because:
$$Ch(E_m\boxdot E_m)(x)=\sum_{\lambda}|(E_m\boxdot E_n)[m.n]/S_{\lambda_1}\times S_{\lambda_2}\times\dots| \, m_{\lambda}(x)$$
$Ch$ being the Frobenius character. An analogous result is obtained for the arithmetic product of elementary symmetric functions, taking the arithmetic product of the sign representations $\Lambda_m$ and $\Lambda_n$. Maybe this can be helpful for expressing their arithmetic product in terms of themselves.