DescriptionIn this 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. After 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 the analogue of the theory for compact groups. 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 Statistics. The book of Diaconis [5] is an excellent starting point. Other applications: There are many more applications. One can see the book of Terras [4] for some more. 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. Pre-requisites
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