On plethysms and Sylow branching coefficients

Abstract

We prove a recursive formula for plethysm coefficients of the form $a_{\lambda ,(m)}^{\mu }$, 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.

arXiv comment35 pages, 6 figures; further explanation and detail added throughout, in particular in Section 5.1, results unchanged; typos corrected; acknowledgements updated