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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 $\Lambda_n$.