Representation Theory of finite groups (Reciprocity Laws)

Chhillar, Pankaj and Kumar, Narasimha (2017) Representation Theory of finite groups (Reciprocity Laws). Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

I have greatly benefited reading a book of G. James and M. Liebeck for Representa- tion Theory and Kenneth Ireland and Michael Rosen for Classical introduction to modern number theory. I have presented a detailed version of the main results in this book.I have divided this thesis in two parts one part contains representation theory of finite groups and second part contains classical introduction to modern number theory. The main intersting thing in Representation Theory is to study a group G as a group of matrices. First we define FG -modules, group algebra , FG - Homomorphism. Then there is a big result namely Maschke’s Theorem, this tells that for a finite group G , if U be the FG -submodule of the FG-module then the FG- module can be written as the direct sum of two of its FG-submodule.And the valid- ity of Maschke’s Theorem hold is for the field of complex and real numbers.After this i discussed about the FG -homomomorphism with a big result namely Schur’s Lemma, For irreducible C G -modules , the C G -homomorphism is either a C G - isomorphism or it is the zero map and if we have a C G -homomorphism from the C G -module to itself then it is the scalar multiple of the identity endomorphism. Then i work on Characters of the C G -modules,inner product of characters and the number of the irreducible character with the proff that, all the irreducible charac- ters of a group G form a basis of the vector space of all class function on G . Then i discuss about how we can restrict a C G -module to a subgroup H of G and then how to construct a C G -module from a given C H -module.Then i discussed some topics in classical number theory including introduction to unique factorization in Z and congrunces, quadratic reciprocity. Then some results using the Legen- dre Symbol , Gauss’ Lemma. Then i worked on the Law of Quadratic Reciprocity, Quadratic Gauss sum , Jacobi sums and related results and then the law of Cubic and Biquadratic reciprocity. And then talked about the Affine spaces, Projective spaces, polynomial over finte field.

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