# 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?

• 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
• 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
• @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

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.

• 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

I've been looking into this and here are two references that I think answer this question.

The characters of the symmetric group can be obtained from those of the full linear group in a similar manner to that used for the orthogonal group, namely by considering a tensor corresponding to any partition $(\lambda)$ of any integer n, and removing all possible contractions with the fundamental forms (2, p. 392). The remainder when all contractions are removed is an irreducible character, provided that $n-p > \lambda_1$, and it is not difficult to see that it is in fact the character of the symmetric group corresponding to the partition $(n-p, \lambda_1,..., \lambda_i)$. It is convenient to represent by $[\lambda]$ not this character, but the corresponding S-function $$[\lambda] = \{n-p,\lambda_1,...,\lambda_i\}$$

The symmetric groups are thus treated quite differently from the linear and other continuous groups: the orthogonal, rotation and symplectic groups. The characters of the groups are know in terms of S functions and the usual method of calculating such things as Kronecker products of the representations of these groups is to use S-functional expressions for their characters and the powerful algebra of S functions associated with the n-independent outer product rule. The labels that arise from this approach are the same as those that arise from tensorial arguments. The aim of this paper is to show that the symmetric groups, $\Sigma_n$, may be treated in an n-independent manner similar to that used for the restriced groups $O_n$ and $Sp_n$, rather than in the usual n-dependent manner requiring a development of the somewhat complicated algebra of inner products of S functions.

After reading these references (and perhaps a few others, but I found these to be the most explicit) I would say the answer to your question "Is this fact already in the literature?" is a qualified 'yes.' It seems some mathematicians knew that characters of the symmetric group could be realized similar to S-functions and orthogonal/symplectic characters, but I don't think they are particularly explicit (and even Littlewood seems to equate the character with the S function).

Rosa Orellana and I posted a paper in 2015 (Symmetric group characters as symmetric functions, https://arxiv.org/abs/1605.06672) that found (exactly as you and Sami Assaf did) the symmetric group realized as permutation matrices has characters that are symmetric functions in the eigenvalues of the permutation matrices. King and Butler provide tables of these characters in terms of Schur functions up to n=5 for the embedding of $S_{n}$ in $Gl_{n-1}$. This is equivalent to providing the Schur expansion of the characters we were calling ${\tilde s}_\lambda[X+1]$.