Identities in character tables of $S_n$

TL;DR

This paper explores numerous identities within the character tables of symmetric groups $S_n$, demonstrating the use of computer algebra systems to discover and prove complex mathematical relationships beyond manual capabilities.

Contribution

It shows that character tables of $S_n$ exhibit many identities similar to Pascal's triangle and introduces a Maple package for automating the discovery and proof of these identities.

Findings

Numerous identities in $S_n$ character tables identified.

Computer algebra system effectively proves deep identities.

Provides a Maple package for further exploration.

Abstract

In the classic "Concrete Math", by Graham, Patashnik and Knuth, it is stated that "The numbers in Pascal's triangle satisfy, practically speaking, infinitely many identities, so it is not too surprising that we can find some surprising relationships by looking closely." The aim of this note is to indicate that a similar statement seems to hold for the character tables of the symmetric groups $S_n$. Just as important, it is a case-study in using a computer algebra system to prove deep identities, way beyond the ability of mere humans. This article is accomanied by a Maple pacgage, Sn, and ample output, avaialble from the webpage http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/sn.html .