16
$\begingroup$

The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.

Is it true that for $n\gg 0$, the row of the character table of $S_n$ corresponding to the sign representation has the most negative entries of any row (about half of them)?

Note that for $S_4$, in addition to the sign representation row there are two other rows which also have 2 negative entries.

Possibly the fact that row sums are positive is relevant here.

Improve this question
$\endgroup$
5
  • $\begingroup$ Do we count each permutation once or each conjugacy class once? $\endgroup$
    – Fedor Petrov
    Apr 26, 2022 at 20:22
  • $\begingroup$ @FedorPetrov: each conjugacy class. The character table is a $p(n)\times p(n)$ table, where $p(n)$ is the number of partitions of $n$. $\endgroup$
    – Sam Hopkins
    Apr 26, 2022 at 20:24
  • 1
    $\begingroup$ Some questions of a somewhat similar flavor that have been considered lately include whether most entries in the character table are highly divisible, or in fact are equal to $0$ (see e.g. arxiv.org/abs/2010.12410). $\endgroup$
    – Sam Hopkins
    Apr 26, 2022 at 20:28
  • $\begingroup$ Is it clear/known that "about half of conjugacy classes have negative signature" ? $\endgroup$
    – Denis Serre
    May 2, 2022 at 9:22
  • 2
    $\begingroup$ @DenisSerre: the sum of the row corresponding to the sign representation is oeis.org/A000700. This quantity grows much slower than $p(n)$, hence why I said “about half.” $\endgroup$
    – Sam Hopkins
    May 2, 2022 at 10:38

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.