Weyl Group Multiple Dirichlet Series

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Weyl Group Multiple Dirichlet Series

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Author : Ben Brubaker
language : en
Publisher: Princeton University Press
Release Date : 2011-07-05

Weyl Group Multiple Dirichlet Series written by Ben Brubaker and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-07-05 with Mathematics categories.

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

Weyl Group Multiple Dirichlet Series

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Author : Ben Brubaker
language : en
Publisher:
Release Date : 1940

Weyl Group Multiple Dirichlet Series written by Ben Brubaker and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1940 with Dirichlet series categories.

Residues Of Weyl Group Multiple Dirichlet Series

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Author : Joel B. Mohler
language : en
Publisher: ProQuest
Release Date : 2009

Residues Of Weyl Group Multiple Dirichlet Series written by Joel B. Mohler and has been published by ProQuest this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.

We give explicit computations of a pair of double Dirichlet series first studied by the Friedberg, Hoffstein, and Lieman. These computations are performed in the rational function field because, in this context, the series are power series which turn out to be rational functions.

Multiple Dirichlet Series Automorphic Forms And Analytic Number Theory

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Author : Solomon Friedberg
language : en
Publisher: American Mathematical Soc.
Release Date : 2006

Multiple Dirichlet Series Automorphic Forms And Analytic Number Theory written by Solomon Friedberg and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.

Multiple Dirichlet series are Dirichlet series in several complex variables. A multiple Dirichlet series is said to be perfect if it satisfies a finite group of functional equations and has meromorphic continuation everywhere. The earliest examples came from Mellin transforms of metaplectic Eisenstein series and have been intensively studied over the last twenty years. More recently, many other examples have been discovered and it appears that all the classical theorems on moments of $L$-functions as well as the conjectures (such as those predicted by random matrix theory) can now be obtained via the theory of multiple Dirichlet series. Furthermore, new results, not obtainable by other methods, are just coming to light. This volume offers an account of some of the major research to date and the opportunities for the future. It includes an exposition of the main results in the theory of multiple Dirichlet series, and papers on moments of zeta- and $L$-functions, on new examples of multiple Dirichlet

Crystal Bases Representations And Combinatorics

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Author : Daniel Bump
language : en
Publisher: World Scientific Publishing Company
Release Date : 2017-01-17

Crystal Bases Representations And Combinatorics written by Daniel Bump and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-17 with Mathematics categories.

This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.

Twisted Weyl Group Multiple Dirichlet Series Over The Rational Function Field

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Author : Holley Ann Friedlander
language : en
Publisher:
Release Date : 2013

Twisted Weyl Group Multiple Dirichlet Series Over The Rational Function Field written by Holley Ann Friedlander and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Dirichlet series categories.

Let K be a global field. For each prime p of K, the p-part of a multiple Dirichlet series defined over K is a generating function in several variables for the p-power coefficients. Let _ be an irreducible, reduced root system, and let n be an integer greater than 1. Fix a prime power q 2 Z congruent to 1 modulo 2n, and let Fq(T) be the field of rational functions in T over the finite field Fq of order q. In this thesis, we examine the relationship between Weyl group multiple Dirichlet series over K = Fq(T) and their p-parts, which we define using the Chinta-Gunnells method [10]. Our main result shows that Weyl group multiple Dirichlet series of type _ over Fq(T) may be written as the finite sum of their p-parts (after a certain variable change), with "multiplicities" that are character sums. This result gives an analogy between twisted Weyl group multiple Dirichlet series over the rational function field and characters of representations of semi-simple complex Lie algebras associated to _. Because the p-parts and global series are closely related, the result above follows from a series of local results concerning the p-parts. In particular, we give an explicit recurrence relation on the coefficients of the p-parts, which allows us to extend the results of Chinta, Friedberg, and Gunnells [9] to all _ and n. Additionally, we show that the p-parts of Chinta and Gunnells [10] agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [4, 5] (in the cases when both constructions apply).

Eisenstein Series And Applications

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Author : Wee Teck Gan
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-12-22

Eisenstein Series And Applications written by Wee Teck Gan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-12-22 with Mathematics categories.

Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. Bringing together contributions from areas which do not usually interact with each other, this volume introduces diverse users of Eisenstein series to a variety of important applications. With this juxtaposition of perspectives, the reader obtains deeper insights into the arithmetic of Eisenstein series. The central theme of the exposition focuses on the common structural properties of Eisenstein series occurring in many related applications.

Multiple Dirichlet Series L Functions And Automorphic Forms

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Author : Daniel Bump
language : en
Publisher: Springer
Release Date : 2012-07-09

Multiple Dirichlet Series L Functions And Automorphic Forms written by Daniel Bump and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-07-09 with Mathematics categories.

Multiple Dirichlet Series, L-functions and Automorphic Forms gives the latest advances in the rapidly developing subject of Multiple Dirichlet Series, an area with origins in the theory of automorphic forms that exhibits surprising and deep connections to crystal graphs and mathematical physics. As such, it represents a new way in which areas including number theory, combinatorics, statistical mechanics, and quantum groups are seen to fit together. The volume also includes papers on automorphic forms and L-functions and related number-theoretic topics. This volume will be a valuable resource for graduate students and researchers in number theory, combinatorics, representation theory, mathematical physics, and special functions. Contributors: J. Beineke, B. Brubaker, D. Bump, G. Chinta, G. Cornelissen, C.A. Diaconu, S. Frechette, S. Friedberg, P. Garrett, D. Goldfeld, P.E. Gunnells, B. Heim, J. Hundley, D. Ivanov, Y. Komori, A.V. Kontorovich, O. Lorscheid, K. Matsumoto, P.J. McNamara, S.J. Patterson, M. Suzuki, H. Tsumura.

Automorphic Forms And L Functions I

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Author : David Ginzburg
language : en
Publisher: American Mathematical Soc.
Release Date : 2009

Automorphic Forms And L Functions I written by David Ginzburg and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.

Includes articles that represent global aspects of automorphic forms. This book covers topics such as: the trace formula; functoriality; representations of reductive groups over local fields; the relative trace formula and periods of automorphic forms; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions.

Number Theory Analysis And Geometry

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Author : Dorian Goldfeld
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-12-21

Number Theory Analysis And Geometry written by Dorian Goldfeld and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-21 with Mathematics categories.

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas of the field, namely Number Theory, Analysis, and Geometry, representing Lang’s own breadth of interest and impact. A special introduction by John Tate includes a brief and fascinating account of the Serge Lang’s life. This volume's group of 6 editors are also highly prominent mathematicians and were close to Serge Lang, both academically and personally. The volume is suitable to research mathematicians in the areas of Number Theory, Analysis, and Geometry.