前辅文
1 Topics in representation theory of finite groups T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli
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 μ from ({ωn},{αn})
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
2.6 Concepts of independence and graph products
2.6.1 From classical to commutative independence
2.6.2 Graph products
2.6.3 Central Limit Theorem for Cartesian powers
2.6.4 Monotone independence and comb product
2.6.5 Boolean independence and star product
2.6.6 Convolutions of spectral distributions
2.6.7 Notes
References
3 Laplacian eigenvalues and optimality R. A. Bailey and P. J. Cameron
3.1 Block designs in experiments
3.1.1 Experiments in blocks
3.1.2 Complete-block designs
3.1.3 Incomplete-block designs
3.1.4 Matrix formulae
3.1.5 Eigenspaces of real symmetric matrices
3.1.6 Fisher’s Inequality
3.1.7 Constructions
3.1.8 Partially balanced designs
3.1.9 Laplacian matrix and information matrix
3.1.10 Estimation and variance
3.1.11 Reparametrization
3.1.12 Exercises
3.2 Laplacian matrices and their eigenvalues
3.2.1 Which graph is best?
3.2.2 Graph terminology
3.2.3 The Laplacian of a graph
3.2.4 Isoperimetric number
3.2.5 Signed incidence matrix
3.2.6 Generalized inverse; Moore–Penrose inverse
3.2.7 Electrical networks
3.2.8 The Matrix-Tree Theorem
3.2.9 Markov chains
3.2.10 Exercises
3.3 Designs, graphs and optimality
3.3.1 Two graphs associated with a block design
3.3.2 Laplacian matrices
3.3.3 Estimation and variance
3.3.4 Resistance distance
3.3.5 Spanning trees
3.3.6 Measures of optimality
3.3.7 Some optimal designs
3.3.8 Designs with very low replication
3.3.9 Exercises
3.4 Further topics
3.4.1 Sylvester designs
3.4.2 Sparse versus dense
3.4.3 Variance-balanced designs
3.4.4 Recognising a concurrence graph
3.4.5 Other graph parameters
3.4.6 Some open problems
3.4.7 Exercises
References
4 Symbolic dynamics and the stable algebra of matrices M. Boyle and S. Schmieding
4.1 Overview
4.2 Basics
4.2.1 Topological dynamics
4.2.2 Symbolic dynamics
4.2.3 Edge SFTs
4.2.4 The continuous shift-commuting maps
4.2.5 Powers of an edge SFT
4.2.6 Periodic points and nonzero spectrum
4.2.7 Classification of SFTs
4.2.8 Strong shift equivalence of matrices, classification of SFTs
4.2.9 Shift equivalence
4.2.10 Williams’ shift equivalence conjecture
4.2.11 Appendix 2
4.3 Shift equivalence and strong shift equivalence over a ring
4.3.1 SE-Z+: dynamical meaning and reduction to SE-Z
4.3.2 Strong shift equivalence of matrices over a ring
4.3.3 SE, SSE and det(I?tA)
4.3.4 Shift equivalence over a ring R
4.3.5 SIM-Z and SE-Z: some example classes
4.3.6 SE-Z via direct limits
4.3.7 SE-Z via polynomials
4.3.8 Cokernel of (I?tA), a Z[t]-module
4.3.9 Other rings for other systems
4.3.10 The module-theoretic formulation of SE over a ring
4.3.11 Appendix 3
4.4 Polynomial matrices
4.4.1 Background
4.4.2 Presenting SFTs with polynomial matrices
4.4.3 Algebraic invariants in the polynomial setting
4.4.4 Polynomial matrices: from elementary equivalence to conjugate SFTs
4.4.5 Classification of SFTs by positive equivalence in I?NZC
4.4.6 Functoriality: flow equivalence in the polynomial setting
4.4.7 Appendix 4
4.5 Inverse problems for nonnegative matrices
4.5.1 The NIEP
4.5.2 Stable variants of the NIEP
4.5.3 Primitive matrices
4.5.4 Irreducible matrices
4.5.5 Nonnegative matrices
4.5.6 The Spectral Conjecture
4.5.7 Boyle–Handelman Theorem
4.5.8 The Kim–Ormes–Roush Theorem
4.5.9 Status of the Spectral Conjecture
4.5.10 Laffey’s Theorem
4.5.11 The Generalized Spectral Conjectures
4.5.12 Appendix 5
4.6 A brief introduction to algebraic K-theory
4.6.1 K1 of a ring R
4.6.2 NK1(R)
4.6.3 Nil0(R)
4.6.4 K2 of a ring R
4.6.5 Appendix 6
4.7 The algebraic K-theoretic characterization of the refinement of strong shift equivalence over a ring by shift equivalence
4.7.1 Comparing shift equivalence and strong shift equivalence over a ring
4.7.2 The Algebraic Shift Equivalence Problem
4.7.3 Strong shift equivalence and elementary equivalence
4.7.4 The refinement of shift equivalence over a ring by strong shift equivalence
4.7.5 The SE and SSE relations in the context of endomorphisms
4.7.6 Appendix 7
4.8 Automorphisms of SFTs
4.8.1 Simple automorphisms
4.8.2 The center of Aut(σA)
4.8.3 Representations of Aut(σA)
4.8.4 Dimension representation
4.8.5 Periodic point representation
4.8.6 Inerts and the sign-gyration compatibility condition
4.8.7 Actions on finite subsystems
4.8.8 Notable problems regarding Aut(σA)
4.8.9 The stabilized automorphism group
4.8.10 Mapping class groups of subshifts
4.8.11 Appendix 8
4.9 Wagoner’s strong shift equivalence complex, and applications
4.9.1 Wagoner’s SSE complexes
4.9.2 Homotopy groups for Wagoner’s complexes and Aut(σA)
4.9.3 Counterexamples to Williams’ conjecture
4.9.4 Kim–Roush relative sign-gyration method
4.9.5 Wagoner’s K2-valued obstruction map
4.9.6 Some remarks and open problems
4.9.7 Appendix 9
References
Subject Index
Author Index
