Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains
Cambridge University Press, Mar 6, 2008 - Mathematics
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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abelian action of G algebra automorphism bijection bound lemma clearly column commutative conﬁguration conjugate convolution Corollary corresponding decomposition deduce defined Deﬁnition denote eigenspace eigenvalue element equivalent ergodic Example Exercise exists finite group ﬁrst follows Fourier transform g G G G L(G G L(X G Xn G-invariant G-orbits Gelfand pair given graph group G integer irreducible representations isomorphic Lemma Let G linear operator Markov chain Moreover multiplicity notation Note orbits orthogonality relations orthonormal basis partition permutation representation poset probability measure Proof Proposition prove random walk recall representation of G respect scalar product Section Show simple random walk spherical functions spherical poset spherical representation stochastic matrix subgroup subsets subspace Suppose symmetric group Theorem transition matrix transpositions trivial unitary vector space vertex vertices Young diagram
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Page 11 - P(B|A) is called the conditional probability of the event B given that the event A has occurred.