Where do stable Kronecker coefficients live "in nature"?

Question

Background:

For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric group $S_{|\lambda|}$. Even if $\lambda[N]$ is not a partition, we can make sense of $\chi_{\lambda[N]}$ as a class function of $S_N$ using a determinant, and Murnaghan proved that there exist coefficients $G^\nu_{\lambda, \mu}$ (the stable, or reduced, Kronecker coefficients) such that

$\chi_{\lambda[N]} \chi_{\mu[N]} = \sum_\nu G^\nu_{\lambda, \mu} \chi_{\nu[N]}$

for all $N \ge 0$.

Question:

In a sense, the construction of $\lambda[N]$ is a bit ad hoc. Is there a more representation-theoretic way to define these coefficients? In particular, if $|\lambda| + |\mu| = |\nu|$, then $G^\nu_{\lambda, \mu}$ coincides with the corresponding Littlewood-Richardson coefficient, so I am hopeful there is some connection. I am looking for an answer which addresses the following point: as I have defined these coefficients, it seems that the most accessible way to work with these coefficients is to use combinatorics. If I wanted to use tools from say, invariant theory or algebraic geometry, what is a more natural context for these coefficients to appear?

Answer 1

Just for the record, I would like to confirm that David Jordan's feeling was right. Deligne's category $Rep(S_t)$ is monoidal, depends on a parameter $t$ and is semisimple for $t\not\in\mathbb N$. In that case, its simple objects $X_\lambda$ are parameterized by partions $\lambda$ of arbitrary size and the multipicity of $X_\nu$ in $X_\lambda\otimes X_\mu$ is precisely the stable Kronecker coefficient $G_{\lambda,\mu}^\nu$.

The idea is as follows: For $t=N\in\mathbb N$ there is a monoidal functor $F:Rep(S_t)\to Rep(S_N)$ (the latter is the ordinary category of $S_N$-representations). If $N$ is big enough (with respect to a fixed $\lambda$), the simple object $X_\lambda$ is also defined for $t=N$. Deligne shows that $F(X_\lambda)$ is the irreducible $S_N$-representation $V_{\lambda[N]}$ corresponding the partition $\lambda[N]$. Therefore, $X_\lambda\otimes X_\mu$ has the same decomposition as $F(X_\lambda\otimes X_\mu)=V_{\lambda[N]}\otimes V_{\mu[N]}$ yielding the result. For details see Inna Aizenbud's paper Deligne categories and reduced Kronecker coefficients.