A classification of Breuil modules

Let p be a prime number and f a positive integer with f < p. In this paper, we determine the structure of simple Breuil modules of type ⊕i=1nωfki\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \oplus _{i = 1}^n\omega _f^{{k_i}}$$\end{document} corresponding to n-dimensional irreducible representations of GQp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_{{{\bf{Q}}_p}}}$$\end{document}. We also describe the extensions of those simple Breuil modules if they correspond to mod p reductions of strongly divisible modules that correspond to Galois stable lattices in potentially semistable representations of GQp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_{{{\bf{Q}}_p}}}$$\end{document} with Hodge-Tate weights {0, 1, …, n − 1} and Galois type ⊕i=1nω˜fki\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \oplus _{i = 1}^n\widetilde\omega _f^{{k_i}}$$\end{document}.