Title: Weyl group multiple Dirichlet series : Type A combinatorial theory /
Authors: Brubaker, Ben, 1976-; Bump, Daniel, 1952-; Friedberg, Solomon, 1958-
Publisher: Princeton University Press,
Year: 2011
Subjects: Dirichlet series.; Weyl groups.
Abstract: 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.
Contents: 1. Type A Weyl group multiple dirichlet series -- 2. Crystals and Gelfand -Tsetlin patterns -- 3. Duality -- 4. Whittaker functions -- 5. Tokuyama's theorem -- 6. Outline of the proof -- 7. Statement B implies statement A -- 8. Cartoons -- 9. Snakes -- 10. Noncritical resonances -- 11. Types -- 12. Knowability -- 13. The reduction to statement D -- 14. Statement E implies statement D -- 15. Evaluation of Ar and A and statement G -- 16. Concurrence -- 17. Conclusion of the proof -- 18. Statement B and crystal graphs -- 19. Statement B and the Yang-Baxter equation -- 20. Crystals and p-adic integration.