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Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have $$\ell(\pi) = |\{1 \leq i < j \leq n \mid \pi(i) > \pi(j)\}|.$$ Furthermore, we know that $$\sum_{\pi \in S_n} sign(\pi) \ell(\pi) = 0$$ for all $n \geq 4$. So I was wondering: What about other characters or class functions? Given a class function $f \in Cl_{\mathbb{Q}}(S_n)$, what can we say about $$\sum_{\pi \in S_n} f(\pi)\ell(\pi)\,\,\, ?$$ What about the case where $f$ is an irreducible character? Given that the length is already known for a long time, I was rather sure that this problem has already been solved. However, I couldn't even find $$\sum_{\pi \in S_n} \ell(\pi) fix(\pi) = \frac{n!n(n-1)}{4} - \frac{(n+1)!}{6}$$ in the OEIS, where $fix(\pi)$ denotes the number of fixed points of $\pi$ (and yes, the two summands there are exactly the sum for the two irreducible characters that make up $fix$).

So, after this long introduction, my question is: Did someone already consider this question and am I just not able to find their work? Does anyone here know of a paper mentioning or even solving this problem, is there an application of such a solution in the study of the Coxeter length or other combinatorial functions? Can you suggest any paper, author, book,... where I might find something about this topic, is there maybe a standard textbook on combinatorial representation theory?

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  • $\begingroup$ I think the standard reference for the combinatorial representation theory of the symmetry group is Bruce Sagan's "The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions." I don't know if it has anything to say about your question. $\endgroup$ – Nathan Reading Jun 26 '18 at 13:29
  • $\begingroup$ I wonder if it would be more helpful to study the polynomials $\sum_{\pi\in S_n}f(\pi)q^{\ell(\pi)}$ for class functions $f$. These are nice certainly for the trivial character and for the sign character and it wouldn't surprise me if they are nice (and known) for other characters. The numbers you are asking for are obtained by differentiating and then setting $q=1$ $\endgroup$ – Nathan Reading Jun 28 '18 at 20:22
  • $\begingroup$ @NathanReading That sounds interesting, thanks, I will try to see if anything turns out there. :) Also thanks a lot for recommending the book, it is really nice to read and already gave me quite a few ideas how to further study my problem. $\endgroup$ – Dirk Jul 3 '18 at 9:49
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A big thanks to Axel Hultman, who sent me an email to tell me that he basically answered my question in his 2014 paper Permutation statistics of products of random permutations (theorem 6.2).
Once again I'm really amazed by the fast and good feedback you get here on MathOverflow.
Should anyone be interested in this topic or know of related research, feel free to leave me a message or send me an email, I'm always open for discussion.

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