co.combinatorics - Where do stable Kronecker coefficients live "in nature"?

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?