Weyl group multiple Dirichlet series : type A combinatorial theory

Author(s)

• Brubaker, Ben
• Bump, Daniel
• Friedberg, Solomon

Bibliographic Information

Weyl group multiple Dirichlet series : type A combinatorial theory

Ben Brubaker, Daniel Bump, and Solomon Friedberg

（Annals of mathematics studies, no. 175）

Princeton University Press, 2011

• : hardcover
• : pbk

Note

Includes bibliographical references (p. [143]-147) and index

Description and Table of Contents

Volume

: hardcover ISBN 9780691150659

Description

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.

Volume

: pbk ISBN 9780691150666

Description

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.

Table of Contents

*FrontMatter, pg. i*Contents, pg. v*Preface, pg. vii*Chapter One. Type A Weyl Group Multiple Dirichlet Series, pg. 1*Chapter Two. Crystals and Gelfand-Tsetlin Patterns, pg. 10*Chapter Three. Duality, pg. 22*Chapter Four. Whittaker Functions, pg. 26*Chapter Five. Tokuyama's Theorem, pg. 31*Chapter Six. Outline of the Proof, pg. 36*Chapter Seven. Statement B Implies Statement A, pg. 51*Chapter Eight. Cartoons, pg. 54*Chapter Nine. Snakes, pg. 58*Chapter Ten. Noncritical Resonances, pg. 64*Chapter Eleven. Types, pg. 67*Chapter Twelve. Knowability, pg. 74*Chapter Thirteen. The Reduction to Statement D, pg. 77*Chapter Fourteen. Statement E Implies Statement D, pg. 87*Chapter Fifteen. Evaluation of LAMBDAGAMMA and LAMBDADELTA, and Statement G, pg. 89*Chapter Sixteen. Concurrence, pg. 96*Chapter Seventeen. Conclusion of the Proof, pg. 104*Chapter Eighteen. Statement B and Crystal Graphs, pg. 108*Chapter Nineteen. Statement B and the Yang-Baxter Equation, pg. 115*Chapter Twenty. Crystals and p-adic Integration, pg. 132*Bibliography, pg. 143*Notation, pg. 149*Index, pg. 155

by "Nielsen BookData"

Details

• NCID: BB05868290
• ISBN:
  • 9780691150659
  • 9780691150666
• LCCN: 2010037073
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Princeton, N.J.
• Pages/Volumes: xii, 158 p.
• Size: 25 cm
• Classification:
  • LCC : QA295
  • DC22 : 515/.243
• Subject Headings:
  • LCSH : Dirichlet series
  • LCSH : Weyl groups
• Parent Bibliography ID:
  • BA00300336