# 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?

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I wonder if it could be related to the "representation theory" coming from the tensor category $Rep S_\lambda$ defined by Deligne. I am thinking in particular about when N goes to infinity. It's only a vague hope which is why I leave it as a comment rather than an answer. Also, do you mean $N>>0$ rather than $N\geq 0$? – David Jordan Mar 24 '10 at 21:13
Argh, $\lambda$ as the subscript of $S$ was a terrible choice. Replace that with $S_\nu$... – David Jordan Mar 24 '10 at 21:14
No, it's valid for $N \ge 0$. We only need $N \gg 0$ for all of the subscripts in question to be partitions. – Steven Sam Mar 24 '10 at 23:47
Pak & Panova have recently put a few papers on arxiv treating Kronecker coefficients (focusing on complexity questions, etc.) They are still mysterious (they are not known to count lattice points inside some nice polytopes, which LR-coeffs are known to do). However, I suggest to have a look at these papers. – Per Alexandersson Aug 8 '14 at 9:22