On weight elimination for ${\mathrm{G}\mathrm{L}}_n({\mathbb{Q}}_{p^f})$

We show that the modular Serre weights of a sufficiently generic mod $p$ Galois representation of an unramified $p$-adic field are themselves generic, and give precise bounds on the genericity, by extending previous work of Emerton, Gee and Herzig. Our bounds are nearly optimal in some cases. We use this to improve recent weight elimination theorems.