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arXiv · 2110.14552

On plethysms and Sylow branching coefficients

Stacey Law·Yuji Okitani
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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.

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Stacey Law, Yuji Okitani. 2021-10-27. On plethysms and Sylow branching coefficients. https://arxiv.org/abs/2110.14552

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