On a positivity conjecture in the character table of $S_n$

TL;DR

This paper investigates a conjecture about the Schur-positivity of sums of power sums over certain partitions in the character table of the symmetric group, confirming it in specific cases and raising new questions.

Contribution

The paper proves the conjecture for specific classes of partitions and introduces new questions on Schur positivity in the context of symmetric group characters.

Findings

Confirmed the conjecture for partitions in specific intervals.

Identified new Schur positivity questions.

Provided partial results supporting the conjecture.

Abstract

In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $\mu\in [(n-4,1^4), (n)]$ or $\mu\in [(1^n), (3,1^{n-3})], $ or $\mu=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$ Many new Schur positivity questions are presented.