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Let $\lambda$ be a partition of $n$ and $\chi^\lambda$ be the character of $S_n$ associated to it. Given any conjugacy class $C$, I want to prove that $$\lambda\mapsto \frac{\chi^\lambda(C)}{f^\lambda}=\frac{\chi^\lambda(C)}{\chi^\lambda(\text{id})}=\frac{\chi^\lambda(C)}{\dim\chi^\lambda}$$ is a polynomial.

This was affirmed without proof in [1] (Proposition 2.9). The article refers to [2] for a proof. In [2] the proof is not straightforward at all (at least for a undergraduate). I think that, since we want to prove something weaker it should be easier.

[1] Eskin, Alex, and Andrei Okounkov. "Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials." Inventiones Mathematicae 145.1 (2001): 59-103.

[2] Kerov, Serguei. "Polynomial functions on the set of Young diagrams." CR Acad. Sci. Paris Sér. I Math. 319 (1994): 121-126.

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  • $\begingroup$ I don't see your claim in arxiv.org/abs/math/0006171 . Is the published version that much different? $\endgroup$ Mar 11, 2019 at 20:09
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    $\begingroup$ Oh! Are you saying that $\chi^\lambda\left(C\right) / \dim \chi^\lambda$ is a polynomial in $\lambda_1, \lambda_2, \ldots, \lambda_n$ where $\lambda$ is ranging over partitions of length $\leq n$ ? I think I am used to a different notion of "polynomial functions of partitions" (or maybe not so different but just differently defined). $\endgroup$ Mar 11, 2019 at 20:11
  • $\begingroup$ @darijgrinberg Yes! I am saying that it is polynomial in $\lambda_1,\dotsc,\lambda_n$. $\endgroup$
    – Gabriel
    Mar 11, 2019 at 20:22
  • $\begingroup$ It makes no sense to fix $C$. How does $C$ vary with $\lambda$? For instance, do we fix the number of cycles of length $\geq 2$ and let the number of fixed points vary? $\endgroup$ Mar 12, 2019 at 14:51
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    $\begingroup$ Is Theorem 2.6 in Valentin Féray, Stanley’s formula for characters of the symmetric group, Ann. Comb., 13(4) : 453--461 what you are looking for? (I am a bit confused by the various things being equated.) If so, this still leaves the question of finding it proven anywhere, as for some reason the reference is missing. $\endgroup$ Mar 12, 2019 at 17:26

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