  We look at various questions related to filtrations in $p$-adic Hodgetheory,
using a blend of building and Tannakian tools. Specifically,Fontaine and
Rapoport used a theorem of Laffaille on filtered isocrystalsto establish a
converse of Mazur's inequality for isocrystals. Wegeneralize both results to
the setting of (filtered) $G$-isocrystalsand also establish an analog of
Totaro's $\otimes$-product theoremfor the Harder-Narasimhan filtration of
Fargues.
