There Are 718 6-Point Topologies, Quasi-Orderings, and Transgraphs
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Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Adams Operations and Symmetries of Representation Categories Arxiv
Adams operations and symmetries of representation categories Ehud Meir and Markus Szymik May 2019 Abstract: Adams operations are the natural transformations of the representation ring func- tor on the category of finite groups, and they are one way to describe the usual λ–ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, deter- mines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and to show that monoidal equivalences that preserve the second Adams operations do not have to be symmetric. Along the way, we classify all possible symmetries and all monoidal auto- equivalences of representation categories of finite groups. MSC: 18D10, 19A22, 20C15 Keywords: Representation rings, Adams operations, λ–rings, symmetric monoidal cate- gories 1 Introduction Every finite group G can be reconstructed from the category Rep(G) of its finite-dimensional representations if one considers this category as a symmetric monoidal category. This follows from more general results of Deligne [DM82, Prop. 2.8], [Del90]. If one considers the repre- sentation category Rep(G) as a monoidal category alone, without its canonical symmetry, then it does not determine the group G. See Davydov [Dav01] and Etingof–Gelaki [EG01] for such arXiv:1704.03389v3 [math.RT] 3 Jun 2019 isocategorical groups. Examples go back to Fischer [Fis88]. The representation ring R(G) of a finite group G is a λ–ring. -
H&Uctures and Representation Rings of Compact Connected Lie Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA ELsEMER Journal of Pure and Applied Algebra 121 (1997) 69-93 h&uctures and representation rings of compact connected Lie groups Akimou Osse UniversitC de Neuchrftel I Mathkmatiques, Rue Emile Argand II, Ch-2007 Neuchatel, Switzerland Communicated by E.M. Friedlander; received 10 November 1995 Abstract We first give an intrinsic characterization of the I-rings which are representation rings of compact connected Lie groups. Then we show that the representation ring of a compact connected Lie group G, with its I-structure, determines G up to a direct factor which is a product of groups Sp(Z) or SO(21+ 1). This result is used to show that a compact connected Lie group is determined by its classifying space; different (and independent) proofs of this result have been given by Zabrodsky-Harper, Notbohm and Msller. Our technique also yields a new proof of a result of Jackowski, McClure and Oliver on self-homotopy equivalences of classifying spaces. @ 1997 Elsevier Science B.V. 1991 Math. Subj. Class.: 55R35, 55N15, 57TlO 1. Introduction Fix a compact connected Lie group G and let R(G) denote the free abelian group generated by the isomorphism classes of complex irreducible representations of G. With the multiplication induced by tensor products of representations, R(G) becomes a commutative ring with unit; it is called the (complex) representation ring of G. The correspondence G --+ R(G) is fknctorial, so that R(G) is an invariant of the Lie group G. -
OF the AMERICAN MATHEMATICAL SOCIETY 157 Notices February 2019 of the American Mathematical Society
ISSN 0002-9920 (print) ISSN 1088-9477 (online) Notices ofof the American MathematicalMathematical Society February 2019 Volume 66, Number 2 THE NEXT INTRODUCING GENERATION FUND Photo by Steve Schneider/JMM Steve Photo by The Next Generation Fund is a new endowment at the AMS that exclusively supports programs for doctoral and postdoctoral scholars. It will assist rising mathematicians each year at modest but impactful levels, with funding for travel grants, collaboration support, mentoring, and more. Want to learn more? Visit www.ams.org/nextgen THANK YOU AMS Development Offi ce 401.455.4111 [email protected] A WORD FROM... Robin Wilson, Notices Associate Editor In this issue of the Notices, we reflect on the sacrifices and accomplishments made by generations of African Americans to the mathematical sciences. This year marks the 100th birthday of David Blackwell, who was born in Illinois in 1919 and went on to become the first Black professor at the University of California at Berkeley and one of America’s greatest statisticians. Six years after Blackwell was born, in 1925, Frank Elbert Cox was to become the first Black mathematician when he earned his PhD from Cornell University, and eighteen years later, in 1943, Euphemia Lofton Haynes would become the first Black woman to earn a mathematics PhD. By the late 1960s, there were close to 70 Black men and women with PhDs in mathematics. However, this first generation of Black mathematicians was forced to overcome many obstacles. As a Black researcher in America, segregation in the South and de facto segregation elsewhere provided little access to research universities and made it difficult to even participate in professional societies. -
The Representation Ring of the Structure Group of the Relative Frobenius Morphism
THE REPRESENTATION RING OF THE STRUCTURE GROUP OF THE RELATIVE FROBENIUS MORPHISM MARKUS SEVERITT Dissertation zur Erlangung des Doktorgrads der Fakult¨atf¨urMathematik Universit¨atBielefeld vorgelegt von Markus Severitt Juni 2010 Gedruckt auf alterungsbest¨andigemPapier˚˚ISO 9706 THE REPRESENTATION RING OF G(n; r) 1 Contents Introduction 2 Organization 3 Acknowledgements 4 1. Basic Notions and Results for Algebraic Groups 5 1.1. Representations 7 1.2. Lie Algebras 9 1.3. Quotients 12 1.4. The Frobenius Morphisms 14 2. Basics About the Algebraic Group G(n; r) 19 2.1. Two Conditions 20 2.2. Important Subgroups 24 2.3. Weight Spaces 28 2.4. The Lie Algebra 29 3. Triangulated Groups 34 3.1. Pretriangulations and Triangulated Morphisms 34 3.2. Triangulations and Irreducible Representations 36 4. Representations of Reductive Groups 41 4.1. Irreducible Representations 41 4.2. Irreducible Representations of Frobenius Kernels 46 4.3. The Representation Ring of GLn 47 5. r-Triangulated Groups with Reductive Hearts 53 5.1. r-Triangulations 53 5.2. Reductive Hearts 55 6. Transfer Homomorphisms 59 6.1. First Type 59 6.2. Second Type 59 6.3. Third Type 62 6.4. Relations 64 7. Differentials and Cartier's Theorem 68 8. Irreducible G(n; r)-Representations 75 8.1. Lie Algebra Action on I(V ) 76 8.2. Fundamental Weights 79 8.3. The Final Case 80 9. The Representation Ring of G(n; r) 87 References 93 2 MARKUS SEVERITT Introduction Let k be a field of prime characteristic p. For a smooth k-variety X of dimension n, the r-th relative Frobenius morphism r (r) FX : X ! X is an fppf-fiber bundle with fibers pr pr R(n; r) := k[x1; : : : ; xn]=(x1 ; : : : ; xn ) over k. -
Robert Macpherson Seen Through the Eyes of Others
Robert MacPherson seen through the eyes of others Compiled by Lizhen Ji June 1, 2006 Happy Birthday, Bob! 1 Introduction Robert MacPherson has made fundamental contributions to many branches of mathematics. His best known result is probably the joint work with Mark Goresky which established the intersection (co)homology of singular spaces.1 Another major piece of work, joint with Bill Fulton, also concerns the inter- section theory, which is rooted in a problem of Hilbert and led to the award2 winning book by Bill Fulton. 3 A partial list of other major topics in his work is the following: 1. Stratified Morse theory4 2. Characteristic classes of singular spaces, Rimann-Roch, K-groups etc. 3. Topological trace formula and weighted cohomology 1See Steven Kleiman’s survey paper for the discovery and many applications of the intersection homology, The development of intersection homology theory, in A century of mathematics in America, Part II, 543–585, Hist. Math., 2, Amer. Math. Soc., Providence, RI, 1989. 21996 Steele Prize of the American Mathematical Society. 3Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Springer, 1998. 4See the survey article by David Massey in the special issues of Pure and Applied Mathematics Quarterly. 1 4. Geometric approach to Langlands program (equivariant cohomology etc) 5. Reduction theory and geometry of locally symmetric varieties5 6. Kalai conjecture on IH of toric varieties 7. Combinatorial manifolds 8. Configuration spaces and compactifications of various spaces 9. Polylogarithms, etc His approach to mathematics is geometric.6 In fact, his lectures are also geometric. Pictures and diagrams in colors are an important part. -
Maa 5229 Introductory Analysis 2 Change(S) Requested
UGPC APPROVAL __________________ Florida Atlantic University UFS APPROVAL ___________________ SCNS SUBMITTAL _________________ CONFIRMED ______________________ BANNER POSTED ___________________ CATALOG POSTED __________________ Graduate Programs—COURSE CHANGE REQUEST WEB POSTED _____________________ DEPARTMENT NAME: COLLEGE OF: MATHEMATICAL SCIENCES SCIENCE COURSE PREFIX & NUMBER: CURRENT COURSE TITLE: MAA 5229 INTRODUCTORY ANALYSIS 2 CHANGE(S) REQUESTED SHOW “X” IN FRONT OF OPTION SHOW “X” IN FRONT OF OPTION CHANGE PREFIX FROM TO: CHANGE TITLE TO: CHANGE COURSE NO. FROM TO: CHANGE DESCRIPTION TO: CHANGE CREDITS FROM TO: CHANGE PREREQUISITES TO: Continuation of topics in MAA 5228. Metric space topology, uniform convergence, Arzela-Ascoli theorem, differentiation and CHANGE COREQUISITES TO: integration of single variable functions, power series, Stone- CHANGE OTHER REGISTRATION CONTROLS TO: Weierstrauss Theorem, measure theory, Lebesgue integral, convergence theorems for the Lebesgue integral, CHANGE GRADING FROM TO: absolute continuity, the fundamental theorem of calculus. OTHER CHANGES TO BE EFFECTIVE (TERM): Attach syllabus for ANY FALL 2008 changes to current course information. Will the requested change(s) cause this course to overlap any Any other departments and/or colleges that might be affected by other FAU course(s)? If yes, please list course(s). the change(s) must be consulted. List entities that have been consulted and attach written comments from each. YES NO X NONE TERMINATE COURSE, EFFECTIVE (GIVE LAST TERM COURSE IS TO BE ACTIVE): Faculty Contact, Email, Complete Phone Number: W. KALIES, WKALIES @ FAU . EDU , (561) 297-1107 SIGNATURES SUPPORTING MATERIALS Approved by: Date: Syllabus—must include all criteria as detailed in UGPC Guidelines. Department Chair: __________________________________ _________________ Go to: www. fau.edu/graduate/gpc/index.php College Curriculum Chair: ____________________________ _________________ to access Guidelines and to download this form. -
The Representation Ring and the Center of the Group Ring
The representation ring and the center of the group ring (1) M1 (12) M (123) M2 2 3 2 3 Master's thesis Jos Brakenhoff supervisor Bart de Smit Mathematisch instituut Universiteit Leiden February 28, 2005 Contents 1 Introduction 2 2 Comparison of discriminants 4 2.1 The representation ring and the center of the group ring . 4 2.2 Discriminants . 7 2.3 Divisibility of discriminants . 9 3 Comparison of spectra 12 3.1 Spectra . 12 3.2 The spectrum of the representation ring . 14 3.3 The spectrum of the center of the group ring . 17 4 Comparison of Q-algebras 21 4.1 Q-algebras . 21 4.2 Finite abelian ´etale algebras . 22 4.3 Two categories . 23 4.4 An equivalence of categories . 25 4.5 Brauer equivalence . 31 4.6 Q-algebras, continuation . 32 1 Chapter 1 Introduction Let G be a finite group of order g. For this group we will construct two commutative rings, which we will compare in this thesis. One of these rings, the representation ring, is built up from the representations of G. A representation of the group G is a finite dimensional C-vector space M together with a linear action of G, that is a homomorphism G GL(M). With this action M becomes a C[G]-module. For each represen! tation M of G and an element σ G we look at the trace of the map M M : m σm, which we will denote b2y Tr (σ) If σ and τ are ! 7! M conjugate elements of G, then TrM (σ) = TrM (τ). -
REPRESENTATION THEORY Tammo Tom Dieck
REPRESENTATION THEORY Tammo tom Dieck Mathematisches Institut Georg-August-Universit¨at G¨ottingen Preliminary Version of February 9, 2009 Contents 1 Representations 4 1.1 Basic Definitions . 4 1.2 Group Actions and Permutation Representations . 9 1.3 The Orbit Category . 13 1.4 M¨obius Inversion . 17 1.5 The M¨obius Function . 19 1.6 One-dimensional Representations . 21 1.7 Representations as Modules . 24 1.8 Linear Algebra of Representations . 26 1.9 Semi-simple Representations . 28 1.10 The Regular Representation . 30 2 Characters 34 2.1 Characters . 34 2.2 Orthogonality . 36 2.3 Complex Representations . 39 2.4 Examples . 42 2.5 Real and Complex Representations . 45 3 The Group Algebra 46 3.1 The Theorem of Wedderburn . 46 3.2 The Structure of the Group Algebra . 47 4 Induced Representations 50 4.1 Basic Definitions and Properties . 50 4.2 Restriction to Normal Subgroups . 54 4.3 Monomial Groups . 58 4.4 The Character Ring and the Representation Ring . 60 4.5 Cyclic Induction . 62 4.6 Induction Theorems . 64 4.7 Elementary Abelian Groups . 67 5 The Burnside Ring 69 5.1 The Burnside ring . 69 5.2 Congruences . 73 5.3 Idempotents . 76 5.4 The Mark Homomorphism . 79 5.5 Prime Ideals . 81 Contents 3 5.6 Exterior and Symmetric Powers . 83 5.7 Burnside Ring and Euler Characteristic . 87 5.8 Units and Representations . 88 5.9 Generalized Burnside Groups . 90 6 Groups of Prime Power Order 94 6.1 Permutation Representations . 94 6.2 Basic Examples . 95 6.3 An Induction Theorem for p-Groups . -
The Representation-Ring of a Compact Lie Group
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. GRAEME SEGAL The representation-ring of a compact Lie group Publications mathématiques de l’I.H.É.S., tome 34 (1968), p. 113-128 <http://www.numdam.org/item?id=PMIHES_1968__34__113_0> © Publications mathématiques de l’I.H.É.S., 1968, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE REPRESENTATION RING OF A COMPACT LIE GROUP GRAEME SEGAL INTRODUCTION Let G be a compact group. I shall use the word G-module to mean a finite-dimensional complex vector space M together with a continuous linear action of G on M. If M and N are G-modules, one can form their sum M©N, and with respect to this operation the isomorphism classes of G-modules form an abelian semigroup. The associated abelian group is called R(G): its elements are formal differences [Mg] — [MJ of isomorphism classes of G-modules modulo the equivalence relation generated by [MJ—[MJ^[Mo®N]—[Mi®N]. Because any G-module is isomorphic to a unique sum of simple G-modules, R(G) is the free abelian group on the set G of isomorphism classes of simple G-modules. -
Psh-Algebras and the Shintani Correspondence
PSH-ALGEBRAS AND THE SHINTANI CORRESPONDENCE VICTOR P. SNAITH Contents 1. PSH-algebras over the integers 1 2. The Decomposition Theorem 3 3. The PSH algebra of {GLmFq, m ≥ 0} 3 4. Semi-direct products Gal(Fqn /Fq) ∝ GLtFqn 10 5. R˜ and R00 12 6. Shintani base change 14 7. Counting cuspidals irreducibles of GLnFq 17 −1 T 8. Appendix: An example of w(k∗,∗)Pα,m−α)w(k∗,∗) Pa,m−a 21 References 24 1. PSH-algebras over the integers A PSH-algebra is a connected, positive self-adjoint Hopf algebra over Z. The notion was introduced in [11]. Let R = ⊕n≥0 Rn be an augmented graded ring over Z with multipication m : R ⊗ R −→ R. Suppose also that R is connected, which means that there is an augmentation ring homomorphism of the form =∼ : Z −→ R0 ⊂ R. These maps satisfy associativity and unit conditions. Associativity: m(m ⊗ 1) = m(1 ⊗ m): R ⊗ R ⊗ R −→ R. Unit: ∼ ∼ m(1 ⊗ ) = 1 = m( ⊗ 1); R ⊗ Z = R = Z ⊗ R −→ R ⊗ R −→ R. R is a Hopf algebra if, in addition, there exist comultiplication and counit homomorphisms m∗ : R −→ R ⊗ R Date: 26 February 2016. 1 and ∗ : R −→ Z such that Hopf m∗ is a ring homomorphism with respect to the product (x ⊗ y)(x0 ⊗ y0) = xx0 ⊗ yy0 on R ⊗ R and ∗ is a ring homomorphism restricting to an iso- morphism on R0. The homomorphism m is a coalgebra homomorphism with respect to m∗. The m∗ and ∗ also satisfy Coassociativity: (m∗ ⊗ 1)m∗ = (1 ⊗ m∗)m∗ : R −→ R ⊗ R ⊗ R −→ R ⊗ R ⊗ R Counit: ∼ ∼ m(1 ⊗ ) = 1 = m( ⊗ 1); R ⊗ Z = R = Z ⊗ R −→ R ⊗ R −→ R. -
University of Aberdeen, Scotland, UK Emeritus Prof
CURRICULUM VITÆ DAVID JOHN BENSON Education and Employment Dates Place Degree/Position 2017{ University of Aberdeen, Emeritus Professor Scotland, UK of Mathematics 2005{17 University of Aberdeen, 6th Century Chair Scotland, UK of Pure Mathematics 2000{05 University of Georgia, Distinguished Research Athens GA, USA Professor (NSF grant held) 1994{00 University of Georgia, Full Professor Athens GA, USA (NSF grant held) 1993{94 University of Georgia, Visiting Associate Athens GA, USA Professor 1986{93 University of Oxford, University Lecturer and England, UK fellow of Wolfson College 1984{86 Northwestern University, Assistant Professor Evanston, IL, USA (NSF grant held) 1982{84 Yale University, Gibbs Instructor New Haven, CT, USA (NSF grant held) 1981{82 Arhus˚ Universitet, Danmark Royal Society exchange fellow 1978{81 Trinity College, Cambridge Ph.D. in Mathematics 1977{78 Trinity College, Cambridge Part III Mathematics, honours with distinction 1974{77 Trinity College, Cambridge First class honours in Mathematics tripos 1967{74 Maidstone Grammar School 8 O-levels, 3 A/S-levels (Mathematics A1, Physics A, Chemistry A1) Awards and Honours 2018 Howard Rowlee Lecture, Lincoln, Nebraska, USA 2018 Research Professorship for the MSRI representation theory programme 2017{2020 Reappointed Honorary Visiting Professor at City, University of London 2017 Maurice Auslander Distinguished Lectures, Cape Cod, USA Date: July 2020. 1 2 DAVID JOHN BENSON 2017 Elected Fellow of the Royal Society of Edinburgh (FRSE) 2016 Conference, PIMS, Vancouver, Canada