Tensor Product and Lattice Graphs

Abstract

This thesis is made up of two distinct parts. In Part I we begin by studying graphs in the representation theory of finite groups called McKay graphs. Let F be an algebraically closed field of prime characteristic p. The modular McKay graph of G := SLn(Fp) with respect to its standard FG-module W is a connected, directed graph whose vertices are the irreducible FG-modules and for which there is an edge from a vertex V1 to V2 if V2 occurs as a composition factor of the tensor product of V1 and W. We show that the diameter of this modular McKay graph is 1/2 (p − 1)(n^2 − n). Let V be a simple FGLn(F)-module and let W now denote the standard FGLn(F)-module. The proof of our diameter result uses the existence of certain composition factors of the tensor product of V and W. Motivated to generalise our diameter result and for its own independent importance in the field of modular representation theory we seek to understand composition factors of the tensor product of the wedge square of V and W. Our second main result relates the composition factors of the tensor product of the wedge square of V and W to composition factors of V restricted to a Levi subgroup. We also provide an illustration of how this can be used to prove the existence of composition factors in certain cases. Part II of this thesis is independent of Part I and motivated by number theory, specifically a local-global compatibility result within the Langlands programme known as Breuil’s lattice conjecture. In Part II we study another type of graph which we call lattice graphs. Let E be a finite extension of Qp with ring of integers O and let H := GLn(Fq) where q = p^f . If V is now a simple EG-module then the vertices of the lattice graph corresponding to V are the homothety classes of OG-lattices in V . The directed edges of the lattice graph encapsulate the relationship between the lattices in terms of objects called Serre weights which have importance in the Langlands programme. The third main result in this thesis is the construction of the lattice graphs for certain representations of GL3(Fp).

Date of Award	1 Dec 2022
Original language	English
Awarding Institution	King's College London
Supervisor	Fred Diamond (Supervisor)