Background
For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a partition of $n$.
The irreducible representations of $S_n$ are indexed by partitions of $n$; we denote them by $S_{\lambda}$. The Kronecker coefficients $g_{\lambda \mu}^{\nu}$ are defined by the equality
$$S_{\lambda} \otimes S_{\mu} \cong \bigoplus g_{\lambda \mu}^{\nu} S_{\nu}$$
of $S_n$ representations.
It is a theorem of Murnaghan that $g_{\lambda[n] \mu[n]}^{\nu[n]}$ becomes constant as $n \to \infty$. This constant value is called the stable Kronecker coefficient, and denoted $\overline{g}_{\lambda \mu}^{\nu}$. It is also a result of Murnaghan that, for given $\lambda$ and $\mu$, there are only finitely many $\nu$ for which $\overline{g}_{\lambda \mu}^{\nu} \neq 0$.
Therefore, we can define a commutative, associative ring to be spanned by the generators $\overline{S}_{\lambda}$, with relations
$$\overline{S}_{\lambda} \overline{S}_{\mu} = \sum \overline{g}_{\lambda \mu}^{\nu} \overline{S}_{\nu}.$$
I'll call this the stable Kronecker ring.
Question
I can prove that the stable Kronecker ring is isomorphic to the ring of symmetric functions. Is this fact already in the literature?
