DOCSLIB.ORG

Thesis Submitted in Fulfilment of the Requirements for the Degree of Doctor of Philosophy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Thesis Submitted in Fulfilment of the Requirements for the Degree of Doctor of Philosophy A Demazure character formula for the product monomial crystal Joel Gibson September 2020 School of Mathematics and Statistics Faculty of Science The University of Sydney A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy Statement of originality This is to certify that to the best of my knowledge, the content of this thesis is my own work. This thesis has not been submitted for any degree or other purposes. I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged. Signed: Joel Gibson 3 Abstract The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisim- ple simply-laced Lie algebra and a multiset of parameters. The crystal is closely related to the representation theory of truncated shifted Yangians, a family of algebras quantising transversal slices to Schubert varieties in the affine Grassmannian. In this thesis we give a systematic study of the product monomial crystal usingthe novel tool of truncations, resulting in a Demazure-type character formula which is valid in any symmetric bi- partite Kac-Moody type. We establish results on stability of the crystal, and use these and the character formula to show that in type A the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram, as defined by Magyar, Reiner, and Shimozono. 5 Acknowledgements Firstly I would like to express my sincere gratitude to my supervisor Oded Yacobi, for his guidance, encour- agement, enthusiasm, and support throughout my candidature. His mentorship and guidance have helped me navigate the academic world, as well as become a better communicator. I consider myself very fortunate to have had his supervision and mentorship over the past four years. Thanks also go to my associate supervisor An- thony Henderson for some great discussions and direction to valuable references, and Geordie Williamson for a fantastic MSRI workshop and his infectious love of representation theory. I thank all the friends who have supported me over the last four years, both in the faculty and outside. Thanks to Kitty Chang and Joachim Worthington for showing me the ropes as a brand-new PhD student, Alex Casella and Dom Tate for the endless effort they put in to the postgraduate maths community at USyd, and Alex Kerschlfor being an excellent conference companion. The members of the representation theory group we’ve had at Sydney over the last few years have been nothing short of amazing, and particular thanks to Anna Romanov, Yusra Naqvi, and Emily Cliff for fostering such a sense of community here. Also thanks to my peers Joe, Josh, Giulian, Gaston, Kane, and Yee for being a great bunch of people. Between mathematical discussions, postgraduate solidatity, and impromptu performances of Gilbert and Sullivan, they’ve been an endless well of support and encouragement. Finally I’d like to thank my family, without whom none of this would have been possible. My partner Anneka has now seen me through both an Honours thesis and a PhD thesis, and has remained enthusiastic and encouraging through all of the ups and downs of a postgraduate degree. Her constant love, support, patience, and energy has been invaluable, and it is in a large part due to her that this thesis (working title ‘The Crystal Matrix’) is finally complete. My parents have been (as always) supporting and encouraging of me, and their advice has helped me slow down and enjoy myself. Thanks for encouraging and reinforcing both my love of science as well as my love of music — taking both in equal parts over the last four years has been a real treat. Finally, my sister Maddie has always believed in me, always reminded me to be proud of myself, and has been a constant irrepressible source of good vibes and ‘dank’ memes. 7 Contents 1 Introduction 11 1.1 Structure .................................................. 12 2 Notation 13 3 Lie Theory 15 3.1 Cartan data ................................................ 16 3.2 The Coxeter graph ............................................. 17 3.3 The Braid, Weyl, and Cactus groups ................................... 18 3.4 Kac-Moody root data ........................................... 20 3.5 Kac-Moody algebras ........................................... 24 3.6 Quantum groups ............................................. 26 4 Crystals 31 4.1 Crystal bases of integrable modules ................................... 31 4.2 The category of crystals .......................................... 34 4.3 Tensor product of crystals ........................................ 39 4.4 Recognition theorems ........................................... 40 4.5 Weyl group action ............................................. 41 4.6 Cactus group action ............................................ 42 4.7 Addendum ................................................. 43 5 Demazure modules and crystals 45 5.1 Formal Characters ............................................. 45 5.2 Demazure modules and the character formula ............................. 47 5.3 Demazure crystals ............................................. 50 5.4 History: The Demazure character formula ............................... 53 6 Monomial crystals 55 6.1 Nakajima’s Monomial Crystal ...................................... 55 6.2 A variation on the monomial crystal ................................... 60 6.3 The product monomial crystal ...................................... 63 6.4 Labelling elements of the crystal ..................................... 65 6.5 A partial order ............................................... 66 6.6 Supports of monomials .......................................... 67 7 Truncations and the character formula 69 7.1 Truncations defined by upward sets ................................... 69 7.2 A Demazure character formula ...................................... 71 7.3 Truncations are Demazure crystals ................................... 73 8 Nakajima quiver varieties 77 8.1 Representations of quivers ........................................ 77 8.2 Moduli spaces of quiver representations ................................ 78 8.3 Nakajima quiver varieties ......................................... 79 8.4 Vector bundles on quiver varieties .................................... 81 8.5 Graded quiver varieties .......................................... 81 9 Contents 9 Generalised Schur modules 85 9.1 Generalised Specht modules ....................................... 85 9.2 Generalised Schur modules ........................................ 88 9.3 Flagged Schur modules .......................................... 91 9.4 Polynomial characters of GL푛 ...................................... 93 9.5 Type A truncations ............................................ 95 10 Stability of decomposition 99 10.1 Stability of restriction ........................................... 99 10.2 Application: decomposing a product ................................... 102 10.3 Stability of the product monomial crystal for GL푛 ........................... 103 Bibliography 107 10 1 Introduction Let 퐺 be a simply-laced reductive group over the complex numbers ℂ, with Repℂ 퐺 the category of its finite- dimensional algebraic representations. This category is semisimple with simple objects 퐿(휆) indexed by domi- nant weights 휆, and an interesting problem is to define ‘natural’ constructions of these representations forany dominant weight. This problem has been very fruitful, with three such constructions realising 퐿(휆) as: 1. The space of sections Γ(퐺/퐵, ℒ휆) of a line bundle ℒ휆 on the flag variety 퐺/퐵, 2. The cohomology of a Nakajima quiver variety associated to the pair (퐺, 휆), generalising a previous con- struction of Ginzburg using Springer fibres in the case of 퐺 = GL푛, and 3. The intersection homology 퐼퐻(Gr휆) inside the dual affine Grassmannian Gr = 퐺∨((푧))/퐺∨[[푧]] of the spher- ical orbit Gr휆. The fact that these three realisations all give rise to 퐿(휆) might be surprising, since the underlying geometric spaces are rather different, and consequently there has been some progress made to state relationships between these spaces. Throughout the papers [Kam+14; Kam+19a; Kam+19b] the authors investigate the relationship between the second and third realisations, establishing the fact that transverse slices in the dual affine Grass- mannian are symplectic dual to Nakajima quiver varieties. Throughout their study they investigate a family of 휆 휆 non-commutative deformations 푌휇 (R) of the coordinate ring of the transverse slice Gr휇 depending on an integral 휆 휆 set of parameters R, and define a category 풪(푌휇 (R)) of their representations. The sum 풱 (휆, R) = ⨁휇≤휆 풪(푌휇 (R)) carries a categorical (Lie 퐺)-action (in the sense of Chuang and Rouquier [CR04; Rou08]), making the complexi- fied Grothendieck group 푉 (휆, R) = 퐾ℂ(풱 (휆, R)) a representation of 퐺. This thesis concerns the representation 푉 (휆, R) and its crystal ℳ(휆, R), called the product monomial crystal af- ter its embedding into Nakajima’s crystal of monomials. We give a novel new method of analysing the crystal ℳ(휆, R) by certain global truncations, which we use to give a Demazure-type character formula for the crystal, our first main result. Our second main result is specific totype A, where we show that the product monomial crystal is in fact the crystal of a previously-studied family of modules called the generalised Schur modules for column- convex
Load more

Recommended publications
  • Alexandre Grothendieck Eléments De Biographie
    Alexandre Grothendieck Eléments de biographie « Ce qui vient au monde pour ne rien troubler ne mérite ni égards ni patience ». René Char, Fureur et mystère Le 14 novembre 2014, les journaux, les radios, les télévisions du monde entier ont annoncé la mort, à Saint-Girons, en Ariège, d’un mathématicien âgé de 86 ans, au nom imprononçable. Des millions de personnes ont ainsi appris l’existence de cet inconnu au lendemain même de sa disparition. Quant à la communauté mathématique, elle a appris la nouvelle avec émotion, une émotion que Cédric Villani résume par ces mots : « Aucun mathématicien vivant n’était plus révéré de ses pairs que lui. » Pendant trente-cinq ans, tous les ans ou presque, en début d’année, j’ai tenu à dire à mes élèves quelques mots sur ce mathématicien, que je croyais retiré dans une bergerie de l’Hérault. C’est de lui que je vais vous parler, ce soir 1. Alexandre Grothendieck fut l’un des plus grands mathématiciens du 20 e siècle. L’irruption de cet autodidacte dans le monde mathématique, en 1949, ses contributions décisives en analyse fonctionnelle et en géométrie algébrique au cours de vingt années de création ininterrompue, sa rupture spectaculaire avec le monde mathématique, en 1970, et sa longue retraite, ont fait de lui un personnage mythique : Grothendieck fut un génie solitaire au destin romanesque, comme la science pure aime à en produire, de temps en temps, depuis trois millénaires : Evariste Galois, Albert Einstein, Srinivasa Ramanujan, Ettore Majorana, Alan Turing, Stephen Hawking, Grigori Perelman… Solitaire,
    [Show full text]
  • Fundamental Algebraic Geometry
    http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
    [Show full text]
  • 1:34 Pm April 11, 2013 Red.Tex
    1:34 p.m. April 11, 2013 Red.tex The structure of reductive groups Bill Casselman University of British Columbia, Vancouver [email protected] An algebraic group defined over F is an algebraic variety G with group operations specified in algebraic terms. For example, the group GLn is the subvariety of (n + 1) × (n + 1) matrices A 0 0 a with determinant det(A) a = 1. The matrix entries are well behaved functions on the group, here for 1 example a = det− (A). The formulas for matrix multiplication are certainly algebraic, and the inverse of a matrix A is its transpose adjoint times the inverse of its determinant, which are both algebraic. Formally, this means that we are given (a) an F •rational multiplication map G × G −→ G; (b) an F •rational inverse map G −→ G; (c) an identity element—i.e. an F •rational point of G. I’ll look only at affine algebraic groups (as opposed, say, to elliptic curves, which are projective varieties). In this case, the variety G is completely characterized by its affine ring AF [G], and the data above are respectively equivalent to the specification of (a’) an F •homomorphism AF [G] −→ AF [G] ⊗F AF [G]; (b’) an F •involution AF [G] −→ AF [G]; (c’) a distinguished homomorphism AF [G] −→ F . The first map expresses a coordinate in the product in terms of the coordinates of its terms. For example, in the case of GLn it takes xik −→ xij ⊗ xjk . j In addition, these data are subject to the group axioms. I’ll not say anything about the general theory of such groups, but I should say that in practice the specification of an algebraic group is often indirect—as a subgroup or quotient, say, of another simpler one.
    [Show full text]
  • Classification of Reductive Monoid Spaces Over an Arbitrary Field
    Classification of Reductive Monoid Spaces Over an Arbitrary Field Mahir Bilen Can August 16, 2018 Abstract In this semi-expository paper, we review the notion of a spherical space. In particu- lar, we present some recent results of Wedhorn on the classification of spherical spaces over arbitrary fields. As an application, we introduce and classify reductive monoid spaces over an arbitrary field. 1 Introduction The classification of spherical homogenous varieties, more generally of spherical actions, is an important, lively, and very interesting chapter in modern algebraic geometry. It naturally encompasses the classification theories of toric varieties, horospherical va- rieties, symmetric varieties, wonderful compactifications, as well as that of reductive monoids. Our goal in this paper is to give an expository account of some recent work in this field. As far as we are aware, the broadest context in which a classification of such objects is achieved, is in the theory of algebraic spaces. This accomplishment is due to Wedhorn [34]. Here, we will follow Wedhorn’s footsteps closely to derive some conclusions. At the same time, our intention is to provide enough detail to make basic definitions accessible to beginners. We will explain a straightforward application of Wedhorn’s progress to monoid schemes. arXiv:1712.08391v3 [math.AG] 16 Aug 2018 It is not completely wrong to claim that the origins of our story go back to Leg- endre’s work [18], where he analyzed the gravitational potential of a point surrounded by a spherical surface in 3-space. To describe the representative functions of his en- terprise, he found a clever change of coordinates argument and introduced the set of orthogonal polynomials {P (x)} via 1 = P (x)hn, which are now n n 0 √1 2hx+h2 Pn 0 n ≥ − ≥ known as Lagrange polynomials.
    [Show full text]
  • Finite Fields and Complements
    Affine representability results in A1-homotopy theory III: finite fields and complements Aravind Asok∗ Marc Hoyoisy Matthias Wendt Abstract We give a streamlined proof of A1-representability for G-torsors under “isotropic” reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in A1-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups. 1 Introduction/Statement of Results Suppose k is a field. We study torsors under algebraic groups considered in the following definition. Definition 1. If G is a reductive algebraic k-group scheme, we will say that G is “isotropic” if each of the almost k-simple components of the derived group of G contains a k-subgroup scheme isomorphic to Gm. Remark 2. In the above definition, the word “isotropic” is in quotes to distinguish from the standard defi- nition of isotropic for reductive groups, which simply requires the existence of a k-subgroup scheme iso- morphic to Gm (see [Gil10,Definition´ 9.1.1]. Let us spell out what term “isotropic” as defined above means. The derived group of Gder := [G; G] is a semi-simple k-subgroup scheme. As such it has a simply- connected covering group Gsc, which is itself a product of almost k-simple factors. The condition above implies that each such factor is isotropic in the standard sense. 1 Write H (k) for the (unstable) Morel–Voevodsky A -homotopy category [MV99]. Write B G for the usual bar construction of G (which can be thought of as a simplicial presheaf on the category of smooth k- schemes).
    [Show full text]
  • Arxiv:2008.12670V2 [Math.AG] 16 Feb 2021 Upre Npr Yjp AEH Rn Ubr16h03921
    LEFT DEMAZURE-LUSZTIG OPERATORS ON EQUIVARIANT (QUANTUM) COHOMOLOGY AND K-THEORY LEONARDO C. MIHALCEA, HIROSHI NARUSE, AND CHANGJIAN SU Abstract. We study the Demazure-Lusztig operators induced by the left mul- tiplication on partial flag manifolds G/P . We prove that they generate the Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant cohomol- ogy), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way we advertise many properties of the left and right divided difference operators in cohomology and K-theory, and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K- theory, generating Schubert classes, and satisfying a Leibniz rule compatible with the quantum product. 1. Introduction Let G be a complex, semisimple, Lie group, and fix a Borel subgroup B, and a maximal torus T := B ∩ B−, where B− is the opposite Borel subgroup. Fix also P ⊃ B a standard parabolic subgroup. It has been known for some time that the Hecke algebra of the Weyl group W := NG(T )/T , and its degenerations (such as the nil-Hecke algebra), act on the cohomology and K-theory of flag manifolds G/B. What is perhaps less advertised is that each Hecke action is given by operators which come in a left, and a right flavor, with the two flavors playing a complementary role. For the equivariant ∗ cohomology ring HT (G/B), Knutson [Knu03] observed that the source of left/right flavors lies in two different automorphisms of G/B, given by the left and the right multiplication by elements of the Weyl group W .
    [Show full text]
  • Arithmetic Groups (Banff, Alberta, April 14–19, 2013) Edited by Kai-Uwe Bux, Dave Witte Morris, Gopal Prasad, and Andrei Rapinchuk
    Arithmetic Groups (Banff, Alberta, April 14–19, 2013) edited by Kai-Uwe Bux, Dave Witte Morris, Gopal Prasad, and Andrei Rapinchuk Abstract. We present detailed summaries of the talks that were given during a week- long workshop on Arithmetic Groups at the Banff International Research Station in April 2013, organized by Kai-Uwe Bux (University of Bielefeld), Dave Witte Morris (University of Lethbridge), Gopal Prasad (University of Michigan), and Andrei Rapinchuk (University of Virginia). The vast majority of these reports are based on abstracts that were kindly provided by the speakers. Video recordings of lectures marked with the symbol ⊲ are available online at http://www.birs.ca/events/2013/5-day-workshops/13w5019/videos/ Contents 1 1. Introduction and Overview The theory of arithmetic groups deals with groups of matrices whose entries are integers, or more generally, S-integers in a global field. This notion has a long history, going back to the work of Gauss on integral quadratic forms. The modern theory of arithmetic groups retains its close connection to number theory (for example, through the theory of automorphic forms) but also relies on a variety of methods from the theory of algebraic groups, particularly over local and global fields (this area is often referred to as the arithmetic theory of algebraic groups), Lie groups, algebraic geometry, and various aspects of group theory (primarily, homological methods and the theory of profinite groups). At the same time, results about arithmetic groups have numerous applications in differential and hyperbolic geometry (as the fundamental groups of many important manifolds often turn out to be arithmetic), combinatorics (expander graphs), and other areas.
    [Show full text]
  • Recherche ; Direction Générale De La Recherche Et De La Technologie ; Cabinet (1988-1993)
    Recherche ; Direction générale de la recherche et de la technologie ; Cabinet (1988-1993) Répertoire (19990378/1-19990378/41) Archives nationales (France) Pierrefitte-sur-Seine 1999 1 https://www.siv.archives-nationales.culture.gouv.fr/siv/IR/FRAN_IR_019209 Cet instrument de recherche a été encodé en 2012 par l'entreprise Numen dans le cadre du chantier de dématérialisation des instruments de recherche des Archives Nationales sur la base d'une DTD conforme à la DTD EAD (encoded archival description) et créée par le service de dématérialisation des instruments de recherche des Archives Nationales 2 Archives nationales (France) INTRODUCTION Référence 19990378/1-19990378/41 Niveau de description fonds Intitulé Recherche ; Direction générale de la recherche et de la technologie ; Cabinet Date(s) extrême(s) 1988-1993 Nom du producteur • Cabinet du ministre délégué ou du secrétaire d'État chargé de la Recherche ou de l'Enseignement supérieur et de la Recherche Localisation physique Pierrefitte DESCRIPTION Présentation du contenu Sommaire Cabinet d’Hubert Curien, ministre de la Recherche et de la Technologie de 1988 à 1992 ; ministre de la Recherche et de l’Espace de 1992 à 1993 Art 1-41 : Interventions particulières portant notamment sur les déplacements en région et le patronage de manifestations scientifiques du ministre, les remises de prix et de décorations, la tutelle d’organismes de recherche et les relations scientifiques, le financement d’actions de recherche, 1988-1993 TERMES D'INDEXATION région; prix scientifique; organisme de recherche;
    [Show full text]
  • Grothendieck's Bibliography by Topic
    GROTHENDIECK’S BIBLIOGRAPHY BY TOPIC by Giacomo Aiello COMMUNICATIONS TO PARIS ACADEMY OF SCIENCES 1 1. Sur la Complétion du Dual d'un Espace Vectoriel Localement Convexe C. R. Acad. Sc. Paris 230 (1950), 605 – 606 2. Quelques Résultats Relatifs a la Dualité dans les Espaces (F) C. R. Acad. Sci. Paris 230 (1950), 1561 – 1563 3. Critères Généraux de Compacité dans les Espaces Vectoriels Localement Convexes. Pathologie des espaces (LF) C. R. Acad. Sci. Paris 231 (1950), 940 – 941 4. Quelques Résultats sur les Espaces Vectoriels Topologiques C. R. Acad. Sci. Paris 233 (1951), 839 – 841 5. Sur une Notion de Produit Tensoriel Topologique D'espaces Vectoriels Topologiques, et une Classe Remarquable d'Espaces Vectoriels Liéе à Cette Notion С. R. Acad. Sci. Paris 233 (1951), 1556 – 1558 6. Résultats Nouveaux dans la Théorie des Operations Linéaires I C. R. Acad. Sci. Paris 239 (1954), 577 – 579 7. Résultats Nouveaux dans la Théorie des Operations Linéaires II C. R. Acad. Sc. Paris 239 (1954), 607 – 609 1 Since the communications to Paris Academy are not papers in proper sense, we have put them in a special section. FUNCTIONAL ANALYSIS 8. Critères de Compacité dans les Espaces Fonctionnels Généraux Amer. J. Math. 74 (1952), 168 – 186 (December 1950) 9. Sur Certains Espaces de Fonctions Holomorphes I J. Reine Angew. Math. 192 (1952), 35 – 64 (November 1951) 10. Sur Certains Espaces de Fonctions Holomorphes II J. Reine Angew. Math. 192 (1952), 77 – 95 11. Sur les Applications Linéaires Faiblement Compactes d'Espaces du Type C(K) Canadian J. Math. 5 (1953), 129 – 173 (March 1952) 12.
    [Show full text]
  • Bruhat-Tits Theory from Berkovich's Point of View. Analytic Filtrations
    Bruhat-Tits theory from Berkovich’s point of view. Analytic filtrations Arnaud Mayeux To cite this version: Arnaud Mayeux. Bruhat-Tits theory from Berkovich’s point of view. Analytic filtrations. 2020. hal-02566872v1 HAL Id: hal-02566872 https://hal.archives-ouvertes.fr/hal-02566872v1 Preprint submitted on 7 May 2020 (v1), last revised 16 Nov 2020 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bruhat-Tits theory from Berkovich's point of view. Analytic ltrations Arnaud Mayeux Let k be a p-adic eld. We dene ltrations by k-anoid groups, in the Berkovich analytication Gan of a connected reductive group G=k, related to Moy-Prasad ltrations. They are parametrized by a cone, whose basis is the Bruhat-Tits building and whose vertex is the neutral element, in Gan via the notions of Shilov boundary and holomorphically convex envelope. Contents 1 Schematic congruence subgroups 4 2 Berkovich k-analytic spaces 7 2.1 Analytication of k-schemes and k◦-schemes . .7 2.2 Analytic groups and descent . 10 3 Bruhat-Tits buildings and Moy-Prasad ltrations 11 3.1 Rational points . 11 3.2 Demazure group schemes .
    [Show full text]
  • Rational Points on Varieties
    GRADUATE STUDIES IN MATHEMATICS 186 Rational Points on Varieties Bjorn Poonen American Mathematical Society 10.1090/gsm/186 Rational Points on Varieties GRADUATE STUDIES IN MATHEMATICS 186 Rational Points on Varieties Bjorn Poonen American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 14G05; Secondary 11G35. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-186 Library of Congress Cataloging-in-Publication Data Names:Poonen,Bjorn,author. Title: Rational points on varieties / Bjorn Poonen. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Gradu- ate studies in mathematics ; volume 186 | Includes bibliographical references and index. Identifiers: LCCN 2017022803 | ISBN 9781470437732 (alk. paper) Subjects: LCSH: Rational points (Geometry) | Algebraic varieties. | AMS: Algebraic geometry – Arithmetic problems. Diophantine geometry – Rational points. msc | Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields. msc Classification: LCC QA564 .P65 2017 | DDC 516.3/5–dc23 LC record available at https://lccn.loc.gov/2017022803 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service.
    [Show full text]
  • Colored Five-Vertex Models and Demazure Atoms
    COLORED FIVE-VERTEX MODELS AND DEMAZURE ATOMS BEN BRUBAKER, VALENTIN BUCIUMAS, DANIEL BUMP, AND HENRIK P. A. GUSTAFSSON Abstract. Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a biproduct, we will construct Demazure atoms on Kashiwara’s B1 crystal and give new algorithms for computing Lascoux-Schützenberger keys. 1. Introduction Exactly solvable lattice models have found numerous applications in the study of special functions. (See [32, 33, 34, 35, 20, 21, 49, 31] to name but a few.) Here we use the Gelfand school interpretation of “special function,” meaning one that arises as a matrix coefficient of a group representation. If the group is a complex Lie group or a p-adic reductive group, these matrix coefficients include highest weight characters and in particular, Schur polynomials, as well as Demazure characters and various specializations and limits of Macdonald polynomials. Many of these special functions may be studied by methods originating in statistical mechanics, by expressing them as a multivariate generating function (the “partition function”) over the admissible states of a solvable lattice model. The term “solvable” means that the model possesses a solution of the Yang-Baxter equation that often permits one to express the partition function of the model in closed form. Knowing that a special function is expressible as a partition function of a solvable lattice model then leads to a host of interesting combinatorial properties, including branching rules, exchange relations under Hecke operators, Pieri- and Cauchy-type identities, and functional equations.
    [Show full text]