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](http://trac.sagemath.org/ticket/14542).


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](https://mathoverflow.net/questions/120924), where another operation on $\mathbf{Symm}$ is constructed on the power sums first and then happens to be integral for weird reasons.)



Arithmetic product of symmetric functions: why is it integral?
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.


