LEADER 07798nam 2201873Ia 450 001 9910789812403321 005 20210518031856.0 010 $a1-283-01338-X 010 $a9786613013385 010 $a1-4008-3899-1 024 7 $a10.1515/9781400838998 035 $a(CKB)2670000000079628 035 $a(EBL)664641 035 $a(OCoLC)713258718 035 $a(SSID)ssj0000474357 035 $a(PQKBManifestationID)11913292 035 $a(PQKBTitleCode)TC0000474357 035 $a(PQKBWorkID)10453969 035 $a(PQKB)11559210 035 $a(MiAaPQ)EBC664641 035 $a(StDuBDS)EDZ0000406790 035 $a(WaSeSS)Ind00025318 035 $a(DE-B1597)446839 035 $a(OCoLC)979629530 035 $a(DE-B1597)9781400838998 035 $a(Au-PeEL)EBL664641 035 $a(CaPaEBR)ebr10451090 035 $a(CaONFJC)MIL301338 035 $z(PPN)199244618 035 $a(PPN)18795755X 035 $a(EXLCZ)992670000000079628 100 $a20100923d2011 uy 0 101 0 $aeng 135 $aurun#---|uu|u 181 $ctxt 182 $cc 183 $acr 200 10$aWeyl group multiple Dirichlet$b[electronic resource] /$fBen Brubaker, Daniel Bump, and Solomon Friedberg 205 $aCourse Book 210 $aPrinceton, N.J. $cPrinceton University Press$dc2011 215 $a1 online resource (173 p.) 225 1 $aAnnals of mathematics studies ;$vno. 175 300 $aDescription based upon print version of record. 311 0 $a0-691-15065-6 311 0 $a0-691-15066-4 320 $aIncludes bibliographical references and index. 327 $tFront matter --$tContents --$tPreface --$tChapter One. Type A Weyl Group Multiple Dirichlet Series --$tChapter Two. Crystals and Gelfand-Tsetlin Patterns --$tChapter Three. Duality --$tChapter Four. Whittaker Functions --$tChapter Five. Tokuyama's Theorem --$tChapter Six. Outline of the Proof --$tChapter Seven. Statement B Implies Statement A --$tChapter Eight. Cartoons --$tChapter Nine. Snakes --$tChapter Ten. Noncritical Resonances --$tChapter Eleven. Types --$tChapter Twelve. Knowability --$tChapter Thirteen. The Reduction to Statement D --$tChapter Fourteen. Statement E Implies Statement D --$tChapter Fifteen. Evaluation of ?? and ??, and Statement G --$tChapter Sixteen. Concurrence --$tChapter Seventeen. Conclusion of the Proof --$tChapter Eighteen. Statement B and Crystal Graphs --$tChapter Nineteen. Statement B and the Yang-Baxter Equation --$tChapter Twenty. Crystals and p-adic Integration --$tBibliography --$tNotation --$tIndex 330 $aWeyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation. 410 0$aAnnals of mathematics studies ;$vno. 175. 606 $aDirichlet series 606 $aWeyl groups 610 $aBZL pattern. 610 $aClass I. 610 $aEisenstein series. 610 $aEuler product. 610 $aGauss sum. 610 $aGelfand-Tsetlin pattern. 610 $aKashiwara operator. 610 $aKashiwara's crystal. 610 $aKnowability Lemma. 610 $aKostant partition function. 610 $aRiemann zeta function. 610 $aSchur polynomial. 610 $aSchützenberger involution. 610 $aSnake Lemma. 610 $aStatement A. 610 $aStatement B. 610 $aStatement C. 610 $aStatement D. 610 $aStatement E. 610 $aStatement F. 610 $aStatement G. 610 $aTokuyama's Theorem. 610 $aWeyl character formula. 610 $aWeyl denominator. 610 $aWeyl group multiple Dirichlet series. 610 $aWeyl vector. 610 $aWhittaker coefficient. 610 $aWhittaker function. 610 $aYang-Baxter equation. 610 $aYang?Baxter equation. 610 $aaccordion. 610 $aadele group. 610 $aaffine linear transformation. 610 $aanalytic continuation. 610 $aanalytic number theory. 610 $aarchimedean place. 610 $abasis vector. 610 $abijection. 610 $abookkeeping. 610 $abox-circle duality. 610 $aboxing. 610 $acanonical indexings. 610 $acardinality. 610 $acartoon. 610 $acircling. 610 $aclass. 610 $acombinatorial identity. 610 $aconcurrence. 610 $acritical resonance. 610 $acrystal base. 610 $acrystal graph. 610 $acrystal. 610 $adivisibility condition. 610 $adouble sum. 610 $aepisode. 610 $aequivalence relation. 610 $af-packet. 610 $afree abelian group. 610 $afunctional equation. 610 $agenerating function. 610 $aglobal field. 610 $aice-type model. 610 $ainclusion-exclusion. 610 $aindexing. 610 $ainvolution. 610 $aisomorphism. 610 $aknowability. 610 $amaximality. 610 $anodal signature. 610 $anonarchimedean local field. 610 $anoncritical resonance. 610 $anonzero contribution. 610 $ap-adic group. 610 $ap-adic integral. 610 $ap-adic integration. 610 $apartition function. 610 $apolynomial. 610 $apreaccordion. 610 $aprototype. 610 $areduced root system. 610 $arepresentation theory. 610 $aresidue class field. 610 $aresonance. 610 $aresotope. 610 $arow sums. 610 $arow transfer matrix. 610 $ashort pattern. 610 $asix-vertex model. 610 $asnakes. 610 $astatistical mechanics. 610 $asubsignature. 610 $atableaux. 610 $atype. 610 $a?-equivalence class. 610 $a?-swap. 615 0$aDirichlet series. 615 0$aWeyl groups. 676 $a515/.243 700 $aBrubaker$b Ben$f1976-$01468330 701 $aBump$b Daniel$f1952-$056694 701 $aFriedberg$b Solomon$f1958-$01468331 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910789812403321 996 $aWeyl group multiple Dirichlet$93679458 997 $aUNINA