Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations 2008 Edition Contributor(s): Haran, Shai M. J. (Author) |
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ISBN: 3540783784 ISBN-13: 9783540783787 Publisher: Springer OUR PRICE: $47.45 Product Type: Paperback - Other Formats Published: May 2008 |
Additional Information |
BISAC Categories: - Mathematics | Number Theory |
Dewey: 511.42 |
Series: Lecture Notes in Mathematics |
Physical Information: 0.5" H x 6.1" W x 9.2" (0.90 lbs) 222 pages |
Descriptions, Reviews, Etc. |
Publisher Description: In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp, w). The real analogue of the p-adic integers is the interval -1,1], and a probability measure w on it gives rise to a special basis for L2( -1,1], w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of -1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums. |