On the Vertices of Indecomposable Modules over Dihedral 2-groups

On the Tensor Products of Modules for Dihedral 2-Groups

The only groups for which all indecomposable modules are ‘knowable’ are those with cyclic, dihedral, semidihedral, and quaternion Sylow p-subgroups. The structure of the Green ring for groups with cyclic and V4 Sylow p-subgroups are known, but no others have been determined. Of the remaining groups, the dihedral 2-groups have the simplest module category but yet the tensor products of any two i...

Two theorems on the vertices of Specht modules

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module S has non-trivial cyclic vertex if and only if λ has p-weight 1. 2. If p does not divide n and S(n−r,1 ) is indecomposable then its vertex is a p-Sylow subgroup of Sn−r−1 × Sr. Mathematics Subject Classification...

Representations of symmetric groups

In this thesis we study the ordinary and the modular representation theory of the symmetric group. In particular we focus our work on different important open questions in the area. 1. Foulkes’ Conjecture In Chapter 2 we focus our attention on the long standing open problem known as Foulkes’ Conjecture. We use methods from character theory of symmetric groups to determine new information on the...

$PI$-extending modules via nontrivial complex bundles and Abelian endomorphism rings

A module is said to be $PI$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this paper, we focus on direct summands and indecomposable decompositions of $PI$-extending modules. To this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper ...

A Group of Order 8 That’s Hard: Indecomposable String Modules for Dihedral Groups