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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.

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  • $\begingroup$ Do we count each permutation once or each conjugacy class once? $\endgroup$
    – Fedor Petrov
    Commented Apr 26, 2022 at 20:22
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    $\begingroup$ @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
    Commented Apr 26, 2022 at 20:24
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    $\begingroup$ 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
    Commented Apr 26, 2022 at 20:28
  • $\begingroup$ Is it clear/known that "about half of conjugacy classes have negative signature" ? $\endgroup$
    – Denis Serre
    Commented May 2, 2022 at 9:22
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    $\begingroup$ @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
    Commented May 2, 2022 at 10:38

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