Space of Invariant Bilinear Forms Under Representation of a Finite Group

Abstract

For a finite-dimensional vector space V over a field newlineF, the space of invariant bilinear newlineforms under the action of a finite group G on V is newlineconcerned with a representation newlineof G. In this thesis, newlinewe evaluate the various newlinenon-degenerate invariant bilinear newlineforms in the space, and respective dimension of the space. Further for n e N we newlinecount the number of all degree n representations which admits a non-degenerate newlineform. The work of this thesis is highly inspired by the work of many eminent and newlinerenowned mathematicians, few of them are Frobenius, Schur, Burnside, Hurwitz, newlineWeyl, Harish-Chandra, Borel, Stenzel, Pazzis, Artin and Kulkarni etc. newlineThere are 7 chapters representing my work. In chapter 1, quotIntroductionquot, the newlineaim of this thesis with few basic definitions and results of representation theory newlinehas been presented. In Chapter 2Space of Invariant bilinear forms under newlinerepresentation of a group of order 8, we cOunt all degree n representations newlineof a group of order 8. Also discuss the space of invariant bilinear forms under newlinedegree n representation along with its dimension. newline