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 $N \ge 0$.


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?

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.