Orthogonal bases of certain symmetry classes of tensors associated with Brauer characters

Abstract

The main focus of this dissertation is on the existence of an orthogonal basis consisting of standard symmetrized tensors (o-basis for short) of a symmetry class of tensors associated with a Brauer character of a finite group. Most of the work is done for the dihedral group and some results are given for the symmetric group. The existence of an o-basis of a symmetry class of tensors associated with an (ordinary) character of a finite group have been studied by several authors. My study was motivated by the work done on the existence of such a basis of a symmetry class of tensors associated with an (ordinary) irreducible character of a dihedral group.

In Chapter 1 we introduce the basic definitions in character theory. In this a Brauer characters, character of a projective indecomposable module (PI) and a block of a finite group will be introduced. Also in this chapter a generalised orthogonality relation of blocks of a finite group is established. In chapter 2 we introduce the symmetrizer and related notions. Some general results associated with Brauer characters of a finite group will also be given in this chapter. Chapter 3 consists of the results associated with Brauer characters, PIs and blocks of a dihedral group. Finally Chapter 4 lists some result associated with the Brauer characters of the symmetric group.