Irreducible characters of the symmetric group and exponential growth

TL;DR

This paper proves that sequences of irreducible symmetric group characters with Young diagrams bounded linearly grow at least exponentially, providing explicit bounds and insights into their growth behavior.

Contribution

It establishes a lower bound on the exponential growth of certain irreducible symmetric group characters with bounded Young diagrams and computes explicit growth bounds.

Findings

Sequences with bounded Young diagrams grow at least exponentially.

Explicit bounds on the growth rate are derived.

Provides new insights into the asymptotic behavior of symmetric group characters.

Abstract

We consider sequences of degrees of ordinary irreducible $S_n$-characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of $n$ with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound.