# Manchester Number Theory Seminar - Miriam Norris

Dates: | 13 June 2023 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: Miriam Norris (Manchester)

Title: Lattice graphs for representations of GL3(F_p)

Abstract: In recent work Le, Le Hung, Levin and Morra have proved a generalisation of Breuilâ€™s Lattice conjecture in dimension three. This involved showing that lattices inside representations of GL3(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 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 GL3(F_p) allowing us to construct their lattice graphs.

Room: Frank Adams 1