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1 - Topics in representation theory of finite groups
- By Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Edited by R. A. Bailey, University of St Andrews, Scotland, Peter J. Cameron, University of St Andrews, Scotland, Yaokun Wu, Shanghai Jiao Tong University, China
Book:

Groups and Graphs, Designs and Dynamics

Published online:

11 May 2024

Print publication:

30 May 2024, pp 1-86
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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.

Quantum Symmetries of Graph C*-algebras
Simon Schmidt, Moritz Weber
Journal:

Canadian Mathematical Bulletin / Volume 61 / Issue 4 / 01 December 2018

Published online by Cambridge University Press:

20 November 2018, pp. 848-864

Print publication:

01 December 2018
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The study of graph ${{C}^{*}}$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph ${{C}^{*}}$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph ${{C}^{*}}$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Spaces of commuting elements in SU(2)
Part of
- Homotopy theory
M. C. Crabb
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 54 / Issue 1 / February 2011

Published online by Cambridge University Press:

19 January 2011, pp. 67-75
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Based on the work of Adem and Cohen, this note describes an explicit stable decomposition of the space of commuting n-tuples in SU(2) as a wedge of indecomposable summands.