Inertial and Hodge--Tate weights of crystalline representations

Let $K$ be an unramified extension of ${\mathbb{Q}}_p$ and $\rho\colon G_K \rightarrow \operatorname{GL}_n(\overline{$\mathbb{Z}$}p)$ a crystalline representation. If the Hodge–Tate weights of $\rho$ differ by at most $p$ then we show that these weights are contained in a natural collection of weights depending only on the restriction to inertia of $\overline{\rho} = \rho \otimes{\overline{$\mathbb{Z}$}_p} \overline{$\mathbb{F}$}_p$. Our methods involve the study of a full subcategory of $p$-torsion Breuil–Kisin modules which we view as extending Fontaine–Laffaille theory to filtrations of length $p$.