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Background

For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a partition of $n$.

The irreducible representations of $S_n$ are indexed by partitions of $n$; we denote them by $S_{\lambda}$. The Kronecker coefficients $g_{\lambda \mu}^{\nu}$ are defined by the equality $$S_{\lambda} \otimes S_{\mu} \cong \bigoplus g_{\lambda \mu}^{\nu} S_{\nu}$$ of $S_n$ representations.

It is a theorem of Murnaghan that $g_{\lambda[n] \mu[n]}^{\nu[n]}$ becomes constant as $n \to \infty$. This constant value is called the stable Kronecker coefficient, and denoted $\overline{g}_{\lambda \mu}^{\nu}$. It is also a result of Murnaghan that, for given $\lambda$ and $\mu$, there are only finitely many $\nu$ for which $\overline{g}_{\lambda \mu}^{\nu} \neq 0$.

Therefore, we can define a commutative, associative ring to be spanned by the generators $\overline{S}_{\lambda}$, with relations $$\overline{S}_{\lambda} \overline{S}_{\mu} = \sum \overline{g}_{\lambda \mu}^{\nu} \overline{S}_{\nu}.$$

I'll call this the stable Kronecker ring.

Question

I can prove that the stable Kronecker ring is isomorphic to the ring of symmetric functions. Is this fact already in the literature?

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The isomorphism just sends S_{\lambda} to the corresponding Schur function, doesn't it? This should be well-known. –  Qiaochu Yuan Jan 4 '10 at 21:42
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I don't know the answer, but the person I'd ask is Arun Ram who has worked on partition algebras "=" Deligne's S_t "=" stable S_n and is very combinatorial in bent. As far as I can tell his survey with Halverson on partition algebras doesn't mention this result. –  Noah Snyder Jan 4 '10 at 21:45
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@Qiaochu: That isomorphism only works if we take an associated graded of this stable Kronecker ring, where the ring is filtered by the size of the partitions, since $\overline{g}^\nu_{\lambda, \mu} \ne 0$ does not imply that $|\nu| = |\lambda| + |\mu|$. –  Steven Sam Jan 4 '10 at 22:03
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1 Answer

The stable Kronecker ring you defined coincides with Grothendieck ring of Deligne's category Rep(S_t) (for a generic value of t) (see Deligne's paper "La categorie des representations du groupe symetrique S_t, lorsque t n'est pas un entier naturel"). It is proved in Section 5 of this paper that this ring has a natural filtration such that associated graded has Littlewood-Richardson coefficients as its structure constants. This might be related with your isomorphism.

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The existence of this associated graded seems to also follow from the Murnaghan-Littlewood theorem. Section 1 of this paper has a nice explanation and references: arxiv.org/abs/0907.4652 (see Murnaghan theorem and Murnaghan-Littlewood theorem) –  Steven Sam Jan 4 '10 at 21:58
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