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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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  • $\begingroup$ 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 $N\geq 0$? $\endgroup$ – David Jordan Mar 24 '10 at 21:13
  • $\begingroup$ Argh, $\lambda$ as the subscript of $S$ was a terrible choice. Replace that with $S_\nu$... $\endgroup$ – David Jordan Mar 24 '10 at 21:14
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    $\begingroup$ No, it's valid for $N \ge 0$. We only need $N \gg 0$ for all of the subscripts in question to be partitions. $\endgroup$ – Steven Sam Mar 24 '10 at 23:47
  • $\begingroup$ 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. $\endgroup$ – Per Alexandersson Aug 8 '14 at 9:22
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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.

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