On plethysms and Sylow branching coefficients

TL;DR

This paper develops a recursive formula for plethysm coefficients, leading to stability results, resolution of conjectures, and new insights into Sylow branching coefficients for symmetric groups, especially for prime 2.

Contribution

It introduces a recursive formula for plethysm coefficients, proves stability, resolves existing conjectures, and analyzes Sylow branching coefficients for symmetric groups at prime 2.

Findings

Derived a recursive formula for plethysm coefficients.

Proved stability of plethysm coefficients.

Most Sylow branching coefficients for trivial characters are positive.

Abstract

We prove a recursive formula for plethysm coefficients of the form $a^\mu_{\lambda,(m)}$, generalising results on plethysms due to Bruns--Conca--Varbaro and de Boeck--Paget--Wildon. From this we deduce a stability result and resolve two conjectures of de Boeck concerning plethysms, as well as obtain new results on Sylow branching coefficients for symmetric groups for the prime 2. Further, letting $P_n$ denote a Sylow 2-subgroup of $S_n$, we show that almost all Sylow branching coefficients of $S_n$ corresponding to the trivial character of $P_n$ are positive.