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E-raamat: Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains

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, (Università degli Studi di Roma 'La Sapienza', Italy), (Università degli Studi Roma Tre)
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Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov ChainsSuurem pilt
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Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space this book develops the necessary tools for the asymptotic analysis of these processes. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for graduate students and researchers working in the areas of representation theory, group theory, harmonic analysis and Markov chains. Its topics range from the basic theory needed for students new to this area, to advanced topics such as the theory of Green's algebras, the complete analysis of the random matchings, and the representation theory of the symmetric group.

Arvustused

'Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space, this book develops the necessary tools for the asymptotic analysis of these processes.' The Times Higher Education Supplement 'There are not many books that can be used both as an elementary textbook and a research monograph with the same ease and success. This one is a rare example. No prerequisites on probability theory and Markov chains are required; everything is explained in detail. From a researcher's point of view, the introduction and detailed study of Gelfand pairs in the context of finite groups is very valuable. The book can be warmly recommended for anyone interested in the subject and/or looking for interesting applications of representation theory.' EMS Newsletter

Muu info

Self-contained text, ideal for graduate students new to the area and researchers.
Preface xi
Part I Preliminaries, examples and motivations 1
1 Finite Markov chains
3
1.1 Preliminaries and notation
3
1.2 Four basic examples
4
1.3 Markov chains
10
1.4 Convergence to equilibrium
18
1.5 Reversible Markov chains
21
1.6 Graphs
26
1.7 Weighted graphs
28
1.8 Simple random walks
33
1.9 Basic probabilistic inequalities
37
1.10 Lurnpable Markov chains
41
2 Two basic examples on abelian groups
46
2.1 Harmonic analysis on finite cyclic groups
46
2.2 Time to reach stationarity for the simple random walk on the discrete circle
55
2.3 Harmonic analysis on the hypercube
58
2.4 Time to reach stationarity in the Ehrenfest diffusion model
61
2.5 The cutoff phenomenon
66
2.6 Radial harmonic analysis on the circle and the hypercube
69
Part II Representation theory and Gelfand pairs 75
3 Basic representation theory of finite groups
77
3.1 Group actions
77
3.2 Representations, irreducibility and equivalence
83
3.3 Unitary representations
83
3.4 Examples
87
3.5 Intertwiners and Schur's lemma
88
3.6 Matrix coefficients and their orthogonality relations
89
3.7 Characters
92
3.8 More examples
96
3.9 Convolution and the Fourier transform
98
3.10 Fourier analysis of random walks on finite groups
103
3.11 Permutation characters and Burnside's lemma
105
3.12 An application: the enumeration of finite graphs
107
3.13 Wielandt's lemma
110
3.14 Examples and applications to the symmetric group
113
4 Finite Gelfand pairs
117
4.1 The algebra of bi-K-invariant functions
118
4.2 Intertwining operators for permutation representations
120
4.3 Finite Gelfand pairs: definition and examples
123
4.4 A characterization of Gelfand pairs
125
4.5 Spherical functions
127
4.6 The canonical decomposition of L(X) via spherical functions
132
4.7 The spherical Fourier transform
135
4.8 Garsia's theorems
140
4.9 Fourier analysis of an invariant random walk on X
143
5 Distance regular graphs and the Hamming scheme
147
5.1 Harmonic analysis on distance-regular graphs
147
5.2 The discrete circle
161
5.3 The Hamming scheme
162
5.4 The group-theoretical approach to the Hammming scheme
165
6 The Johnson scheme and the Bernoulli–Laplace diffusion model
168
6.1 The Johnson scheme
168
6.2 The Gelfand pair (Sn, Sn-m x Sm) and the associated intertwining functions
176
6.3 Time to reach stationarity for the Bernoulli Laplace diffusion model
180
6.4 Statistical applications
184
6.5 The use of Radon transforms
187
7 The ultrametric space
191
7.1 The rooted tree Tq,n
191
7.2 The group Aut(Tq,n) of automorphisms
192
7.3 The ultrametric space
194
7.4 The decomposition of the space L(Σto the nth) and the spherical functions
196
7.5 Recurrence in finite graphs
199
7.6 A Markov chain on Σto the nth
202
Part III Advanced theory 207
8 Posets and the q-analogs
209
8.1 Generalities on posets
209
8.2 Spherical posets and regular semi-lattices
216
8.3 Spherical representations and spherical functions
222
8.4 Spherical functions via Moebius inversion
229
8.5 q-binomial coefficients and the subspaces of a finite vector space
239
8.6 The q-Johnson scheme
243
8.7 A q-analog of the Hamming scheme
251
8.8 The nonbinary Johnson scheme
257
9 Complements of representation theory
267
9.1 Tensor products
267
9.2 Representations of abelian groups and Pontrjagin duality
274
9.3 The commutant
276
9.4 Permutation representations
279
9.5 The group algebra revisited
283
9.6 An example of a not weakly symmetric Gelfand pair
291
9.7 Real complex and quaternionic representations: the theorem of Frobenius and Schur
294
9.8 Greenhalgebras
304
9.9 Fourier transform of a measure
314
10 Basic representation theory of the symmetric group
319
10.1 Preliminaries on the symmetric group
319
10.2 Partitions arid Young diagrams
322
10.3 Young tableaux and the Specht modules
325
10.4 Representations corresponding to transposed tableaux
333
10.5 Standard tableaux
335
10.6 Computation of a Fourier transform on the symmetric group
343
10.7 Random transpositions
348
10.8 Differential posets
358
10.9 James' intersection kernels theorem
365
11 The Gelfand pair (S2n, S2 Sn) and random matchings
371
11.1 The Gelfand pair (S2n, S2 Sn)
371
11.2 The decomposition of L(X) into irreducible components
373
11.3 Computing the spherical functions
374
11.4 The first nontrivial value of a spherical function
375
11.5 The first nontrivial spherical function
376
11.6 Phylogenetic trees and matchings
378
11.7 Random matchings
380
Appendix 1 The discrete trigonometric transforms 392
Appendix 2 Solutions of the exercises 407
Bibliography 421
Index 433
Tullio Ceccherini-Silberstein is Professor of Mathematical Analysis in the Department of Engineering at the Università del Sannio, Benevento. Fabio Scarabotti is Professor of Mathematical Analysis in the Department of Mathematics at the Università degli Studi di Roma 'La Sapienza'. Filippo Tolli is Assistant Professor of Mathematical Analysis in the Department of Mathematics at the Università Roma Tre.
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