Topics in representation theory of finite groups T. Ceccherini-Silberstein, F. Scarabotti and F. ToUi
1.1 Introduction
1.2 Representation theory and harmonic analysis on finite groups
1.2.1 Representations
1.2.2 Finite Gelfand pairs
1.2.3 Spherical functions
1.2.4 Harmonic analysis of finite Gelfand pairs
1.3 Laplace operators and spectra of random walks on finite graphs
1.3.1 Finite graphs and their spectra
1.3.2 Strongly regular graphs
1.4 Association schemes
1.5 Applications of Gelfand pairs to probability
1.5.1 Markov chains
1.5.2 The Ehrenfest diffusion model
1.6 Induced representations and Mackey theory
1.6.1 Induced representations
1.6.2 Mackey theory
1.6.3 The little group method of Mackey and Wigner
1.6.4 Hecke algebras
1.6.5 Multiplicity-free triples and their spherical functions
1.7 Representation theory of GL(2,Fq)
1.7.1 Finite fields and their characters
1.7.2 Representation theory of the affine group Aff(Fq)
1.7.3 The general linear group GL(2,Fq)
1.7.4 Representations of GL(2,Fq)
References
2 Quantum probability approach to spectral analysis of growing graphs N. Obata
2.1 Introduction
2.2 Basic concepts of quantum probability
2.2.1 Algebraic probability spaces
2.2.2 Spectral distributions
2.2.3 Convergence of random variables
2.2.4 Classical probability vs quantum probability
2.2.5 Notes
2.3 Quantum decomposition
2.3.1 Jacobi coefficients and interacting Fock spaces
2.3.2 Orthogonal polynomials
2.3.3 Quantum decomposition
2.3.4 How to explicitly compute/.t from ({~Z~n}, (an})
2.3.5 Boson, fermion and free Fock spaces
2.3.6 Notes
2.4 Spectral distributions of graphs
2.4.1 Adjacency matrix as a real random variable
2.4.2 IFS structure associated to graphs
2.4.3 Homogeneous trees and Kesten distributions
2.5 Growing graphs
2.5.1 Formulation of question in general
2.5.2 Growing distance-regular graphs
2.5.3 Growing regular graphs
2.5.4 Notes
