DescriptionRepresentation Theory (see also Wikipedia page) is a central area of Pure Mathematics with many applications in Engineering, Statistics and other areas. In this project we will mainly see Representation Theory as an extension of Fourier Theory to situations where the underlying space is not necessarily an abelian group. The focus will be on finite groups hence we will not need to worry about convergence (as for the real line for example) but on the other hand we will consider groups that are not abelian (such as the the symmetric group).In the group project we will start with the basics of Representation Theory following the very nicely written book [1](Chapter 3). We will learn enough of the theory to be able to state and prove the Peter-Weyl theorem in this setting which can also be understood as Fourier Theory on finite groups. In particular the aim is to cover the following topics as part of the group project (with references from [1]):
Group Project
Mode of Operation and Evidence of Learning for the group projectThe project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by presenting the material in their own way, exploring examples (existing or new) and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Individual ProjectAfter that the project can take many directions according to taste. The following is a list of some of the possible directions:Further Representation Theory: One can follow a theoretical direction by learning more about Representatiion Theory. Here are some possible suggestions: the representation theory of the symmetric group (see [1], Chapter 10 for example and many other books on Representation Theory), or Representation of Wreath products of Finite Groups(see [7]) or the analogue of the theory for compact groups, and many more. Gelfand Pairs and Representation Theory: Gelfand Pairs (see also Wikipedia page ) are met in representation theory of groups, but in this project we will be considering only finite Gelfand pairs. Central to the study of Gelfand pairs are so-called spherical functions (Krawtchouk polynomials, Hahn polynomials, etc) and the spherical Fourier transform. These functions, and transforms play a prominent role in many applications (see below). Applications to Error-Correcting Codes: One of the main results in the theory of error-correcting codes is the so-called MacWilliams identities which relate the weight distribution of a code with its dual. The main tool to derive such a relation is a Fourier transform (it is often called Hadamard transform in this setting). A vast generalisation of this can be achieved in the setting of Association Schemes (see also Wikipedia page) which have very interesting algebraic structure, related to Representation Theory. Thanks to the work of Delsarte there is a very interesting connection with the theory of error-correcting codes. (Hamming Scheme, Johnson Scheme, etc). The spherical functions mentioned above are very important here. Random Walks on Distance-regular Graphs: Distance-regular graphs (see also Wikipedia page) are graphs with some nice properties. One can use Fourier analysis to study random walks on such graphs. These random walks model some very interesting diffusion processes such as the Bernoulli-Laplace, and the Ehrenfest diffusion process. The spherical functions mentioned above can be used to study the so-called cutoff phenomena of such models, namely to answer the question, how long does it take to mix things up?. Applications to Statistics:. There are applications of Representation theory to Applied Probability and Statistics. The book of Diaconis [5] is an excellent starting point. Another good starting point with many references is the paper of Hannan [6] i Other applications: There are many more applications. One can see the book of Terras [4] for some more. Mode of Operation and Evidence of Learning for the individual projectThe project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by presenting the material in their own way, exploring examples (existing or new) and theoretical applications of the material, and clearly communicating it in both written and oral formats.Resources
[2] Algebraic Combinatorics I, Association Schemes, E. Banai, T. Ito, The Benjamin/Cummings Publishing Company, 1984. [3] An algebraic approach to the association schemes of coding theory, P. Delsarte, Philips Research Reports, 1973. [4] Fourier analysis on finite groups and applications, A. Terras, CUP 1999. [5] Group Representations in Probability and Statistics , P. Diaconis, IMS Lecture Notes-Monograph Series, 1988. [6] Group Representations and Applied Probability , E.J.Hannan, J. Apple. Prob. 2, 1-68, 1965. [7] Representation Theory and Harmonic Analysis of Wreath products of Finite Groups, T. Ceccherrini-Silberstein, F Scarabotti, F. Tolli, LMS Lecture Notes Series, CUP. Pre-requisites
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