To address the question of whether the sign representation row in the character table of ( S_n ) has the most negative entries for large ( n ), we analyze the behavior of character values and partition lengths.

Key Observations:

1. Sign Representation Character Values: The sign representation's character ( \chi^{\text{sign}} ) evaluated at a conjugacy class ( \mu ) (indexed by a p...

Short answer: Yes. For large n the sign row is the unique one that maximizes the number of negative entries, and that number is exactly the number of odd conjugacy classes (which is asymptotically half of all conjugacy classes). For small n there are a few ties (e.g. n = 4), but these disappear once n is large enough.

Details and why this is the right order of magnitude:

• Conjugacy classes in S...

Yes. In fact one can show that among all irreducible characters of Sₙ the sign‐character is asymptotically “as negative as possible,’’ and in particular for n≫1 it is the unique row of the character table which comes closest to having half its entries negative. Here is a sketch of the proof.

1. Notation and the exact count for the sign–row.
   • Let p(n)=# {partitions of n} be the number of...

Yes. In fact something much stronger is true:
except for the trivial and for the sign representation every irreducible character of Sₙ is zero on “almost all’’ conjugacy classes, so that the number of negative values it can take is negligible compared with the about one-half attained by the sign character. Consequently the sign character is the (unique) row with the largest number of negativ...

This is a very interesting question about the character tables of symmetric groups. Let me analyze what we know and what the question is asking.

The character table of has rows corresponding to irreducible representations (indexed by partitions of ) and columns corresponding to conjugacy classes (also indexed by partitions of ).

Key observations:

1. The sign representation corre...

An excellent and subtle question. The user's hypothesis is that for large , the sign character has the most negative entries in the character table of . Let's analyze this claim.

Summary of the Conclusion

The user's conjecture is plausible and points in the right direction, but is unlikely to be true in its strongest form (that the sign representation is the unique m...

First, recall that the character table of ( S_n ) is based on its irreducible representations, which are indexed by partitions of ( n...

Row of the Character Table of Symmetric Group with Most Negative Entries

This question is about determining which row of the character table of the symmetric group has the most negative entries, particularly for large .

Analysis of the Sign Representation

The sign representation of corresponds to the partition . For a permutation with cycle structure corresponding to ...

At the moment nobody knows a proof (or a counter–example) of the statement

“For all sufficiently large n the sign character has strictly more
negative values than every other irreducible character of S n”.

What is known is the following.

1. Computations.
   • With GAP/MAGMA one can list every irreducible character of S n and count on how many conjugacy classes it is negative.
   ...