Graphs of lattices in representations of finite groups
This thesis work is motivated by the Langlands program, which relates objects from number theory and representation theory. In particular it is motivated by the compatibility of the local and global cases, especially in the mod p case. Given a finite group G, E a finite extension of Qp with ring of integers OE, uniformizer pE, V a finite dimensional E vector space and r G AutE(V an absolutely irreducible representation, we associate to r the following directed graph: its vertices are homothety classes of lattices in V and there is an edge from (the class of) L to L' when pEL L' L and the quotient L/L' is irreducible. We also label the edge according to the isomorphism class of L/L' The central theme of this thesis is the study of stable lattices in p adic representations and their corresponding graphs. In particular, we show certain properties of these associated graphs, including finiteness, connectedness, a duality property and that the length of a cycle is a multiple of the number of the Jordan Holder factors. Moreover, we restrict our attention to certain families of representations arising from admissible representations of GL2 of a p adic field and show further properties of their graphs. In the case where r is of principal series type we compute the bound, that is we find an explicit integer c and the lattice L in V such that all lattices L' in V up to homothety satisfy p EL L' L. In the case where r is of tame principal series type we compute the graphs and investigate their properties. We also compute graphs for certain representations of interest, where most of them are obtained using Magma. The Magma code is attached in the thesis.