1,721,010 research outputs found
Fourier analysis of subgroup conjugacy invariant functions on finite groups
No full textGiven a finite group and a subgroup , we develop a Fourier analysis for -conjugacy invariant functions on , without the assumption that is a multiplicity-free subgroup of . We also study the Fourier transform for functions in the center of the algebra of -conjugacy invariant functions on . We show that a recent calculation of Cesi is indeed a Fourier transform of a function in the center of the algebra of functions on the symmetric group that are conjugacy invariant with respect to a Young subgroup
Harmonic analysis on a finite homogeneous space II: the Gelfand Tsetlin decomposition
No full textIn this paper, we continue the analysis of [28] on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. We extend the theory of Gelfand-Tsetlin bases to permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme. We also extend part of the Okounkov-Vershik theory to the Young permutation module M-a. In particular we constuct explicit Gelfand-Tsetlin bases for the representation S-n-1,S-1. We also give an explicit Gelfand-Tsetlin decomposition for the permutation module associated with a three-parts partitions, using James reformulation of the Young rule by means of intertwining operators (Radon transforms). Several statistical applications, refining previous work by Diaconis, are given. Finally, the spectrum of several invariant operators is determined
Mackey's theory of tau-conjugate representations for finite groups
No full textThe paper contains an exposition of two contributions of Mackey,
together with a more recent result of Kawanaka and Matsuyama, generalized by Bump
and Ginzburg, on the representation theory of a finite group equipped with an involutory
anti-automorphism. Mackey’s first contribution is
a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs.
Mackey’s second contribution is a characterization of simply reducible groups (a notion
introduced by Wigner). The other result is a twisted version of the Frobenius-Schur
theorem, where “twisted” refers to the above-mentioned involutory anti-automorphism.The paper contains an exposition of two contributions of Mackey,
together with a more recent result of Kawanaka and Matsuyama, generalized by Bump
and Ginzburg, on the representation theory of a finite group equipped with an involutory
anti-automorphism. Mackey’s first contribution is
a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs.
Mackey’s second contribution is a characterization of simply reducible groups (a notion
introduced by Wigner). The other result is a twisted version of the Frobenius-Schur
theorem, where “twisted” refers to the above-mentioned involutory anti-automorphism
Spectral analysis of finite Markov chains with spherical symmetries
Get PDFWe generalize the classical Fourier analysis of Gelfand pairs to the setting of groups acting not transitively on a set X. We use this analysis to determine the spectrum of several random walks on graphs. Moreover, as byproduct, we show that, for a new urn diffusion model, the cut-off phenomenon holds
Harmonic analysis of finite lamplighter random walks
No full textRecently, several papers have been devoted to the analysis of lamplighter random walks, in particular, in the case where the underlying graph is the infinite path Z. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the C-2-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. If the graph has a transitive isometry group G, we also describe the spectral analysis in terms of the representation theory of the wreath product C-2 (sic) G. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples have already been studied by Haggstrom and Jonasson by probabilistic methods
Trees, wreath products and finite Gelfand pairs
Get PDFWe 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
Generalized Kaloujnine groups, uniseriality and height of automorphisms
No full textWe show that the Lie action of the Kaloujnine group K(p, n) on the vector space (Fp)(pn) is uniserial. Using some Radon transform techniques we derive a formula for the height of the elements in K(p, n). A generalization of the Kaloujnine groups is introduced by considering automorphisms of a spherically homogeneous tree. We observe that uniseriality fails to hold for these groups and determine their lower central series; finally we discuss in detail Kaloujnine's description of the characteristic subgroups in terms of the (normal) "parallelotopic" subgroups
Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups
No full textThis 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
Mackey's criterion for subgroup restriction of Kronecker products and harmonic analysis on Clifford groups
No full textWe present a criterion for multiplicity-freeness of the decomposition of the
restriction ResG
H(1 ⊗ 2) of the Kronecker product of two generic irreducible represen-
tations 1, 2 of a finite group G with respect to a subgroup H ≤ G. This constitutes a
generalization of a well known criterion due to Mackey (which corresponds to the case
H = G). The corresponding harmonic analysis is illustated by detailed computations on
the Clifford groups G = CL(n), together with the subgroups H = CL(n − 1), for n ≥ 1,
which lead to an explicit decomposition of the restriction of Kronecker products
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