Universal deformation rings and fusion

This thesis is on the representation theory of finite groups. Specifically, it is about finding connections between fusion and universal deformation rings.

Two elements of a subgroup N of a finite group Γ are said to be fused if they are conjugate in Γ, but not in N. The study of fusion arises in trying to relate the local structure of Γ (for example, its subgroups and their embeddings) to the global structure of Γ (for example, its normal subgroups, quotient groups, conjugacy classes). Fusion is also important to understand the representation theory of Γ (for example, through the formula for the induction of a character from N to Γ).

Universal deformation rings of irreducible mod p representations of Γcan be viewed as providing a universal generalization of the Brauer character theory of these mod p representations of Γ.

It is the aim of this thesis to connect fusion to this universal generalization by considering the case when Γ is an extension of a finite group G of order prime to p by an elementary abelian p-group N of rank 2. We obtain a complete answer in the case when G is a dihedral group, and we also consider the case when G is abelian. On the way, we compute for many absolutely irreducible F_pΓ-modules V, the cohomology groups H²(Γ,Hom_Fp(V,V) for i = 1, 2, and also the universal deformation rings R(Γ,V).