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 : 2011
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 2011 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 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.
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).
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
Multiple Dirichlet Series For Affine Weyl Groups
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Author : Ian Whitehead
language : en
Publisher:
Release Date : 2014
Multiple Dirichlet Series For Affine Weyl Groups written by Ian Whitehead and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
The construction is limited to the rational function field, but it also describes the p-part of the multiple Dirichlet series over an arbitrary global field.
Weyl Group Multiple Dirichlet
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Author : Ben Brubaker
language : en
Publisher:
Release Date : 2011
Weyl Group Multiple Dirichlet 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 2011 with Dirichlet series categories.
Recent Trends In Algebraic Combinatorics
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Author : Hélène Barcelo
language : en
Publisher: Springer
Release Date : 2019-01-21
Recent Trends In Algebraic Combinatorics written by Hélène Barcelo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-01-21 with Mathematics categories.
This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.