Orthogonal bases of certain symmetry classes of tensors associated with Brauer characters
Type of Degreedissertation
Mathematics and Statistics
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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.