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Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of $\lambda$ and $\ell(\lambda):=\lambda^t_1$ the length of $\lambda$. Let $\tau\in S_n$ be any transposition. In [R, Theorem 3.5], the following formula is quoted: $$ \binom n2 \cdot \frac{\chi(\tau)}{\chi(1)} = \sum_{i=1}^{\ell(\lambda)} \left( \binom{\lambda_i}2 - \binom{\lambda^t_i}2 \right). $$

I would like to see a proof for this formula. Roichman refers to the paper [I], which itself refers to a German paper by Frobenius, "Über die Charaktere der symmetrischen Gruppe". I was unable to obtain a copy of said paper, but I believe this must somehow follow from the Murnaghan-Nakayama rule (or directly from the Frobenius character formula).

Frankly, I have not invested a lot of time into an attempt to prove this myself because I was hoping that there is a more accessible and detailed reference for such an early formula. I would be very grateful if someone could provide such reference. Of course, if you happen to know a short proof, don't refrain from posting it.

Jesko Hüttenhain
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  • So, for $\lambda = \lambda^t$ we get zero on the RHS. That seems strange. Also, what is $\chi(1)$ supposed to mean? My characters use to have $\chi(1) = 1$. Unless I am misunderstanding something trivial, this formula is very weird... – Marek Feb 08 '13 at 14:33
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    @Marek $\chi(1)$ is the degree of the representation afforded by $\chi$. – Alexander Gruber Feb 08 '13 at 15:26
  • @Alexander: ah, right, thanks. I confused this with characters that are homomorphisms (as opposed to just functions, like here). – Marek Feb 08 '13 at 15:43

1 Answers1

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The paper Über die Charaktere der symmetrischen Gruppe by Frobenius is in the Sitzungsberichte der königlich preußischen Akademie der Wissenschaften zu Berlin of $1900$, available at archive.org in various formats. It begins on p. $516$; the formula you cite seems to correspond to the last formula of the paper on p. $534$. The notation $f^{(\kappa)}=\chi^{(\kappa)}_0$ is introduced on p. $522$, and $h_\varrho=n!/(c(n-c)!)$ is introduced on p. $533$. I'm not sure though how the $a_i$ and $b_i$ are related to your $\lambda_i$ and $\lambda_i^t$. Let me know if you need any help with the German.

joriki
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  • Oh that's great! The German will be the least of my problems. I will check it out right away. – Jesko Hüttenhain Feb 12 '13 at 10:10
  • @Jesko: Sorry, I didn't even check your name, let alone your location :-) – joriki Feb 12 '13 at 11:49
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    As anticipated, this is quite tough to read. Distilling the proof from it would probably require more time than I am willing to invest; Maybe someone knows a reference which is about 100 years younger? Otherwise, I'll accept your answer and just quote the result =). – Jesko Hüttenhain Feb 12 '13 at 14:31
  • Alright then, I shall accept this answer. Thanks a bunch for the research, I am quite happy to know that the Sitzungsberichte can be accessed via arXiv. – Jesko Hüttenhain Feb 18 '13 at 18:47
  • @Jesko: This is archive.org; there's no relation to arxiv.org (other than the name). And you're welcome :-) – joriki Feb 18 '13 at 18:50
  • ooooooohhhhh .... that makes more sense now. xD – Jesko Hüttenhain Feb 18 '13 at 19:36