To address the question of identifying elements ( g \in S_n ) for which there exists an irreducible character ( \chi ) with ( |\chi(g)| > \sqrt{|Z(g)|} e^{-\delta \sqrt{n}} ) (where ( \delta > 0 ) is small), we analyze the structure of permutations, their centralizers, and character bounds from representation theory.

Key Observations:

1. Centralizer Size: For a permutation ( g )...

Write n = ∑k k mk, where mk is the number of k-cycles of g, so |Z(g)| = ∏k (kmk mk!).

Your inequality asks for those g for which there exists some irreducible χ of Sn with |χ(g)| > √|Z(g)| e−δ√n.

There are three simple regimes to keep in mind, which together already give a reasonably sharp picture.

1. What averaging already guarantees (for every g) By column orthogonality, ∑χ |χ(g)|^2 = |Z(g)|,...

The whole problem turns on the size of the centraliser of the permutation
and on the observation that

  ∑χ |χ(g)|² = |Z(g)|  (orthogonality of characters).

Write p(n) for the number of partitions of n ( = the number of irreducible characters of S n ), and fix a constant δ>0. Put

  M(g) := maxχ |χ(g)| .

Of course M(g) ≤ √{|Z(g)|}; the question is when we can have

  M(g) > √{|Z(g)|} eˉ ...

This is an excellent question that touches upon some deep results in the asymptotic representation theory of symmetric groups. Your intuition that such characters exist only when has large cycles is spot on. We can make this intuition precise using a powerful asymptotic formula for the maximum character value on a given conjugacy class.

Summary of the Answer

The condition $|\chi(g...

Here is a fairly complete picture, which shows that nothing “mystical’’ is hiding here – the only time one can beat the naïve central‐​izer bound by even as little as a factor is when has so few cycles that its centralizer is itself of size at most . Equivalently one shows:

1. **(Larsen–Shalev and Féray–Śniady boun...

Throughout let
Z(g)=∏k k^{m_k }m_k ! be the order of the centraliser of the permutation
g=1^{m_1 }2^{m_2 }⋯ in S_n (so ∑k km_k =n) and fix a positive constant δ.
We want to know when there is an irreducible character χ of S_n such that

          |χ(g)| > √Z(g)·e^{-δ√n}.               (1)

1. A very useful elementary observation

The number c(g)=∑k m_k of cycles of g controls ...

Analysis of Large Character Values in the Symmetric Group

The question asks when there exists an irreducible character of such that for a given permutation and small fixed .

Understanding the Condition

This condition is closely related to the formula , which sums over all irreduci...

<think> Okay, the user is asking about large values of characters in the symmetric group. Specifically, they want to know for which elements \( g \in S_n \) there exists an irreducible character \( \chi \) such that \( |\chi(g)| > \sqrt{|Z(g)|} e^{-\delta \sqrt{n}} \) for some small \( \delta > 0 \).

First, I need to recall some basics. For a permutation ( g ) with cycle type ( (m_1, m_2, \ld...

This is an interesting question about character bounds for symmetric groups. Let me analyze what's known and what can be said about your specific bound.

The question asks: for which does there exist an irreducible character such that ?

Key Observations

1. The bound is sharp for regular elements: When is an -cycle, w...