Representation theory and the regular representation

Splitting the regular representation into its group invariant subspaces

Bachelor thesis (2021)

Authors

Q.W.J. van Gulik Electrical Engineering, Mathematics and Computer Science

Contributors

J.G. Spandaw Analysis - EEMCS (mentor)
P.M. Visser Mathematical Physics - EEMCS (mentor)
Dion Gijswijt Discrete Mathematics and Optimization - EEMCS (coach)
Faculty
Electrical Engineering, Mathematics and Computer Science, Electrical Engineering, Mathematics and Computer Science
Copyright: © 2021 Quirijn van Gulik
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Published Date

26-08-2021

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

Representation theory is a branch in mathematics that studies group homomorphisms between a group and the automorphism group of a vector space. A special representation that every group has is the regular representation. This representation permutes all elements of the group in a vector space which dimension is equal to the order of the group.
Within this vector space there are group invariant subspaces. There are several methods to finding representation invariant subspaces of this vector space.

This thesis aims to do two things: First of all, this thesis aims to give the reader an introduction to representation theory, presenting various key concepts, definitions and theorems. Moreover, ways to construct character tables are presented along with multiple worked out examples. Second, the regular representations of D4 and Q8 are decomposed into representation invariant subspaces of C8. To this end, two methods were used.

The first method (A), proposed by dr. Jeroen Spandaw, works out all the possible decompositions of the vector space a regular representation acts on. This is quite a laborious process in which change of basis matrices, expressed in several parameters, will have to be made for all generators of the considered group. The second methods (B), proposed by dr. Paul Visser, is known as the grand orthogonalization method and makes use of an extended version of the character table of the considered group.

Both methods are perfectly fine to make a desired decomposition. However, method A
takes a lot more computational effort than method B to come up with the desired result. The benefit of using method A over method B is that method A considers all possible decompositions, whereas method B only considers one of the infinitely many that are possible.