Positivity of Sylow branching coefficients of symmetric groups

TL;DR

This paper studies the positivity of Sylow branching coefficients in symmetric groups specifically for the prime p=3, completing previous research on odd primes and advancing understanding in this area.

Contribution

It provides a comprehensive analysis of Sylow branching coefficients for p=3, extending prior work and resolving open questions about their positivity.

Findings

Confirmed positivity of Sylow branching coefficients for p=3 in symmetric groups

Extended previous results to cover the case of p=3

Resolved open questions from earlier studies

Abstract

In this article we investigate the positivity of Sylow branching coefficients for symmetric groups when $p = 3$. In particular, we complete the discussion begun by Giannelli and the author in arXiv:1712.02642 (J. Algebra) and developed in arXiv:1909.09446 (J. London Math. Soc.) concerning the case of odd primes.