46 research outputs found
Harmonic analysis of finite lamplighter random walks
Recently, several papers have been devoted to the analysis of lamplighter
random walks, in particular when the underlying graph is the infinite path
. In the present paper, we develop a spectral analysis for
lamplighter random walks on finite graphs. In the general case, we use the
-symmetry to reduce the spectral computations to a series of eigenvalue
problems on the underlying graph. In the case the graph has a transitive
isometry group , we also describe the spectral analysis in terms of the
representation theory of the wreath product . We apply our theory to
the lamplighter random walks on the complete graph and on the discrete circle.
These examples were already studied by Haggstrom and Jonasson by probabilistic
methods.Comment: 29 page
Harmonic analysis on a finite homogeneous space
In this paper, we study harmonic analysis on finite homogeneous spaces whose
associated permutation representation decomposes with multiplicity. After a
careful look at Frobenius reciprocity and transitivity of induction, and the
introduction of three types of spherical functions, we develop a theory of
Gelfand Tsetlin bases for permutation representations. Then we study several
concrete examples on the symmetric groups, generalizing the Gelfand pair of the
Johnson scheme; we also consider statistical and probabilistic applications.
After that, we consider the composition of two permutation representations,
giving a non commutative generalization of the Gelfand pair associated to the
ultrametric space; actually, we study the more general notion of crested
product. Finally, we consider the exponentiation action, generalizing the
decomposition of the Gelfand pair of the Hamming scheme; actually, we study a
more general construction that we call wreath product of permutation
representations, suggested by the study of finite lamplighter random walks. We
give several examples of concrete decompositions of permutation representations
and several explicit 'rules' of decomposition.Comment: 69 page
Discrete Harmonic Analysis. Representations, Number Theory, Expanders and the Fourier Transform
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
Trees, wreath products and finite Gelfand pairs
We present a new construction of finite Gelfand pairs by looking at the action of the full automorphism group of a finite spherically homogeneous rooted tree of type r on the variety V(r, s) of all spherically homogeneous subtrees of type s. This generalizes well-known examples as the finite ultrametric space, the Hamming scheme and the Johnson scheme. We also present further generalizations of these classical examples. The first two are based on Harary's notions of composition and exponentiation of group actions. Finally, the generalized Johnson scheme provides the inductive step for the harmonic analysis of our main construction. (C) 2005 Elsevier Inc. All fights reserved