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This monograph is the first comprehensive treatment of
multiplicity-free induced representations of finite groups as a
generalization of finite Gelfand pairs. Up to now, researchers have
been somehow reluctant to face such a problem in a general
situation, and only partial results were obtained in the
one-dimensional case. Here, for the first time, new interesting and
important results are proved. In particular, after developing a
general theory (including the study of the associated Hecke
algebras and the harmonic analysis of the corresponding spherical
functions), two completely new highly nontrivial and significant
examples (in the setting of linear groups over finite fields) are
examined in full detail. The readership ranges from graduate
students to experienced researchers in Representation Theory and
Harmonic Analysis.
This monograph adopts an operational and functional analytic
approach to the following problem: given a short exact sequence
(group extension) 1 N G H 1 of finite groups, describe the
irreducible representations of G by means of the structure of the
group extension. This problem has attracted many mathematicians,
including I. Schur, A.H. Clifford, and G. Mackey and, more
recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud,
and J.M.G. Fell & R.S. Doran.The main topics are, on the one
hand, Clifford Theory and the Little Group Method (of Mackey and
Wigner) for induced representations, and, on the other hand,
Kirillov's Orbit Method (for step-2 nilpotent groups of odd order)
which establishes a natural and powerful correspondence between Lie
rings and nilpotent groups. As an application, a detailed
description is given of the representation theory of the
alternating groups, of metacyclic, quaternionic, dihedral groups,
and of the (finite) Heisenberg group. The Little Group Method may
be applied if and only if a suitable unitary 2-cocycle (the Mackey
obstruction) is trivial. To overcome this obstacle, (unitary)
projective representations are introduced and corresponding Mackey
and Clifford theories are developed. The commutant of an induced
representation and the relative Hecke algebra is also examined.
Finally, there is a comprehensive exposition of the theory of
projective representations for finite Abelian groups which is
applied to obtain a complete description of the irreducible
representations of finite metabelian groups of odd order.
This self-contained book introduces readers to discrete harmonic
analysis with an emphasis on the Discrete Fourier Transform and the
Fast Fourier Transform on finite groups and finite fields, as well
as their noncommutative versions. It also features applications to
number theory, graph theory, and representation theory of finite
groups. Beginning with elementary material on algebra and number
theory, the book then delves into advanced topics from the
frontiers of current research, including spectral analysis of the
DFT, spectral graph theory and expanders, representation theory of
finite groups and multiplicity-free triples, Tao's uncertainty
principle for cyclic groups, harmonic analysis on GL(2,Fq), and
applications of the Heisenberg group to DFT and FFT. With numerous
examples, figures, and over 160 exercises to aid understanding,
this book will be a valuable reference for graduate students and
researchers in mathematics, engineering, and computer science.
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.
The representation theory of the symmetric groups is a classical
topic that, since the pioneering work of Frobenius, Schur and
Young, has grown into a huge body of theory, with many important
connections to other areas of mathematics and physics. This
self-contained book provides a detailed introduction to the
subject, covering classical topics such as the
Littlewood-Richardson rule and the Schur-Weyl duality. Importantly
the authors also present many recent advances in the area,
including Lassalle's character formulas, the theory of partition
algebras, and an exhaustive exposition of the approach developed by
A. M. Vershik and A. Okounkov. A wealth of examples and exercises
makes this an ideal textbook for graduate students. It will also
serve as a useful reference for more experienced researchers across
a range of areas, including algebra, computer science, statistical
mechanics and theoretical physics.
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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