Surprising Relations Between Sums-Of-Squares of Characters of the Symmetric Group Over Two-Rowed Shapes and Over Hook Shapes

TL;DR

This paper uncovers unexpected relationships between sums of squares of symmetric group characters over two-rowed and hook shapes, revealing new combinatorial identities involving partitions with odd parts and powers of two.

Contribution

It proves novel identities linking character sums over different shapes, extending previous results with human-driven proofs and exploring partitions with odd parts and powers of two.

Findings

Sum over two-rowed shapes with all odd parts equals half the sum over hook shapes with related partitions.

Identifies a pattern where replacing a sequence of powers of two alters the sum in a predictable way.

Provides human proofs of these combinatorial identities, expanding understanding of symmetric group characters.

Abstract

In a recent article (arXiv:1507.03499) (joint with Alon Regev) we studied sums of squares of characters Chi(L,M) of the Symmetric Group over shapes L that are two-rowed, and shapes L that are hook shapes, and M is an arbitrary shape that mostly consists of ones, and designed algorithms for closed-form evaluations of each of these. We noted (and proved) that when M is the shape with n cells consisting of 3 followed by n-3 ones, the former sum equals one half time the analogous sum over hook shapes with n+2 cells and M is the partition consisting of 3,2, followed by n-3 ones. Here we show that this is just a tip of an iceberg, and prove (alas, by purely human means) that the former sum with M consisting of all odd parts, and (possibly) a consecutive string of powers of 2, starting at 2, equals one half of the latter sum where M is replaced by a partition where all the odd parts are retained but the consecutive string of powers of 2: 2,4, ..., $2^{t-1}$ is replaced by $2^t$.