Lattice graphs for representations of $GL_3(\F_p)$

Abstract: In a recent paper Le, Le Hung, Levin and Morra proved a generalisation of Breuil's Lattice conjecture in dimension three. This involved showing that lattices inside representations of $GL_3(\F_p)$ coming from both a global and a local construction coincide. Motivated by this we consider the following graph. For an irreducible representation $\tau$ of a group $G$ over a finite extension $K$ of $\Q_p$ we define a graph on the $\mathcal{O}_K$-lattices inside $\tau$ whose edges encapsulate the relationship between lattices in terms of irreducible modular representations of $G$ (or Serre weights in the context of the paper by Le et al.).

In this talk, I will demonstrate how one can apply the theory of graduated orders and their lattices, established by Zassenhaus and Plesken, to understand the lattice graphs of residually multiplicity free representation over suitably large fields in terms of a matrix called an exponent matrix. Furthermore I will explain how I have been able to show that one can determine the exponent matrices for suitably generic representation go $GL_3(\F_p)$ allowing us to construct their lattice graphs.

number theory

Audience: researchers in the topic

London number theory seminar

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