Weyl Group Multiple Dirichlet Series (eBook)

Type A Combinatorial Theory (AM-175)

Ben Brubaker, Daniel Bump, Solomon Friedberg (Autoren)

eBook Download: EPUB

2011
184 Seiten
Princeton University Press (Verlag)
978-1-4008-3899-8 (ISBN)

Infos

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.

Autor

Ben Brubaker is assistant professor of mathematics at Massachusetts Institute of Technology. Daniel Bump is professor of mathematics at Stanford University. Solomon Friedberg is professor of mathematics at Boston College.

Produkt-Details

• Erscheint lt. Verlag: 5.7.2011
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 168 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Graphentheorie
• Themenwelt: Technik
• Schlagworte: absolute value • Abuse of notation • Accordion • Addition • adele group • affine linear transformation • analytic continuation • Analytic number theory • archimedean place • Basis (linear algebra) • basis vector • Big O notation • bijection • bisection • Bochner's theorem • Boltzmann distribution • bookkeeping • Boundary value problem • box-circle duality • Boxing • BZL pattern • Calculation • Canonical basis • canonical indexings • Cardinality • Cartesian Product • Cartoon • circling • Class • Class I • coefficient • combination • combinatorial identity • combinatorial proof • combinatorics • commutative property • Complex Analysis • Computation • concurrence • Connected component (graph theory) • Contradiction • corollary • critical resonance • Crystal • crystal base • crystal graph • diagram • Diagram (category theory) • Dimension • Dimension (vector space) • Direct proof • Dirichlet series • Disjoint union • divisibility condition • Divisibility rule • double sum • Eigenvalues and Eigenvectors • Eisenstein series • Enumeration • Episode • Equation • _-equivalence class • equivalence class • equivalence relation • Euclidean space • Euler product • Euler's totient function • existential quantification • f-packet • free abelian group • functional equation • Gauss sum • Gelfand-Tsetlin pattern • Generating function • Geometry • global field • ice-type model • inclusion-exclusion • Inclusion–exclusion principle • indexing • Inequality (mathematics) • integrable system • Involute • Involution • isomorphism • Kashiwara operator • Kashiwara's crystal • knowability • Knowability Lemma • Kostant partition function • Langlands dual group • Lattice (group) • L-Function • Lie algebra • Linear combination • Linear map • Mathematical Induction • maximality • maximal torus • Metaplectic Group • Morphism • Natural number • nodal signature • nonarchimedean local field • noncritical resonance • nonzero contribution • Normalizing constant • Notation • ORDER BY • p-adic group • p-adic integral • p-adic integration • Parameter • Parameter (computer programming) • partition function • Partition function (mathematics) • planar graph • polynomial • Polytope • preaccordion • Prototype • Quantity • Quantum group • rational function • Reciprocity law • reduced root system • remainder • Representation Theory • residue class field • resonance • resotope • Riemann zeta function • Right-hand rule • Root of unity • root system • row sums • row transfer matrix • Schtzenberger involution • Schur polynomial • Schützenberger involution • scientific notation • Several Complex Variables • short pattern • six-vertex model • Snake Lemma • Snakes • Special case • Statement A • Statement B • Statement C • Statement (computer science) • Statement D • Statement E • Statement F • Statement G • Statistical Mechanics • SUBGROUP • Subset • subsignature • Summation • _-swap • tableaux • tensor product • Theorem • Tokuyama's Theorem • transfer matrix • Triangular Number • two-dimensional space • Type • Upper and lower bounds • Weyl character formula • Weyl denominator • Weyl Group • Weyl group multiple Dirichlet series • Weyl vector • Whittaker coefficient • Whittaker Function • Yang-Baxter equation • Yang–Baxter equation • YangЂaxter equation • Γ-equivalence class • Γ-swap
• ISBN-10: 1-4008-3899-1 / 1400838991
• ISBN-13: 978-1-4008-3899-8 / 9781400838998