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.
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This book presents an introduction to the representation theory of wreath products of finite groups and harmonic analysis on the corresponding homogeneous spaces. The reader will find a detailed description of the theory of induced representations and Clifford theory, focusing on a general formulation of the little group method. This provides essential tools for the determination of all irreducible representations of wreath products of finite groups. The exposition also includes a detailed harmonic analysis of the finite lamplighter groups, the hyperoctahedral groups, and the wreath product of two symmetric groups. This relies on the generalised Johnson scheme, a new construction of finite Gelfand pairs. The exposition is completely self-contained and accessible to anyone with a basic knowledge of representation theory. Plenty of worked examples and several exercises are provided, making this volume an ideal textbook for graduate students. It also represents a useful reference for more experienced researchers
We present a general introduction to finite Gel'fand pairs and their associated spherical functions yielding different characterizations, examine a few explicit examples, and, for each of these examples, analyze the corresponding probabilistic problem, which will then be solved by applying the general results and the machinery developed for a particular Gel'fand pair.
Let L be an irreducible regular language. Let W be a non-empty set of words (or sub-words) of L and denote by L-W = {v is an element of L:w not subset of v, For Allw is an element of W} the language obtained from L by forbidding all the words w in W. Then the entropy decreases strictly: ent(L-W) < ent(L). In this note we present a new proof of this fact, based on a method of Gromov, which avoids the Perron-Frobenius theory. This result applies to the regular languages of finitely generated free groups and an additional application is presented. (C) 2003 Elsevier B.V. All rights reserved
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