86,570 research outputs found
HARMONIC ANALYSIS ON A FINITE HOMOGENEOUS SPACE
No full textIn 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, we introduce three types of spherical functions. Then we consider the composition of two permutation representations, giving a noncommutative 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
Spectral analysis of finite Markov chains with spherical symmetries
No full textWe 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
Hecke algebras and harmonic analysis on finite groups
No full textLet G be a finite group, K a subgroup and V an irreducible representation
of K. Then the Hecke algebra associated with the triple (G,K,V ) is the commutant of hte representation obtained inducing V from K to G.
Curtis and Fossum derived several
explicit expressions for the characters of Hecke algebras. In the present paper we
give an exposition of their results in the language of finite
harmonic analysis. In particular, we show the connection with the theory of finite
Gelfand pairs
Radon transforms and lamplighter random walks.
No full textWe use a Radon transform approach to compute the spectrum of the lamplighter random walk on the pat
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
Hecke algebras
No full textSia G un gruppo finito e K un sottogruppo. Dalla similarità tra rappresentazioni indotte e rappresentazioni permutazionali abbiamo un isomorfismo tra l'algebra delle funzioni bi-K-invarianti su G e il commutante della rappresentazione ottenuta inducendo la banale da K a G.Let G be a finite group and K ≤ G a subgroup. Recalling the equality between the induced representation (IndKGιK,IndKGC) and the permutation representation (λ, L(G)K), (1.11) yields a ∗-algebra isomorphism between the algebra of bi-K-invariant functions on G and the commutant of the representation obtained by inducing to G the trivial representation of K
- …
