On plethysms and Sylow branching coefficients
Stacey Law, Yuji Okitani
Source abstract
We prove a recursive formula for plethysm coefficients of the form a λ , ( m ) μ , encompassing those which arise in a long-standing conjecture of Foulkes. This also generalises 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.
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