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1 - Topics in representation theory of finite groups

Published online by Cambridge University Press:  11 May 2024

By
Tullio Ceccherini-Silberstein ,
Fabio Scarabotti  and
Filippo Tolli
Edited by
R. A. Bailey ,
Peter J. Cameron  and
Yaokun Wu
R. A. Bailey
Affiliation:
University of St Andrews, Scotland
Peter J. Cameron
Affiliation:
University of St Andrews, Scotland
Yaokun Wu
Affiliation:
Shanghai Jiao Tong University, China
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Summary

This is an introduction to representation theory and harmonic analysis on finite groups. This includes, in particular, Gelfand pairs (with applications to diffusion processes à la Diaconis) and induced representations (focusing on the little group method of Mackey and Wigner). We also discuss Laplace operators and spectral theory of finite regular graphs. In the last part, we present the representation theory of GL(2, Fq), the general linear group of invertible 2 × 2 matrices with coefficients in a finite field with q elements. More precisely, we revisit the classical Gelfand–Graev representation of GL(2, Fq) in terms of the so-called multiplicity-free triples and their associated Hecke algebras. The presentation is not fully self-contained: most of the basic and elementary facts are proved in detail, some others are left as exercises, while, for more advanced results with no proof, precise references are provided.

Keywords

finite groupgroup representationcharacterGelfand pairspherical functionspherical Fourier transformMackey–Wigner little group methodMarkov chainrandom walkEhrenfest diffusion processergodic theoremfinite graphspectral graph theoryLaplace operatordistance-regular graphstrongly regular graphassociation schemefinite fieldaffine group over a finite fieldgeneral linear group over a finite fieldGelfand–Graev representationmultplicity-free tripleHecke algebra
Type
Chapter
Information
Groups and Graphs, Designs and Dynamics , pp. 1 - 86
Publisher: Cambridge University Press
Print publication year: 2024

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