ON LOWER BOUNDS FOR THE DIMENSIONS OF PROJECTIVE MODULES FOR FINITE SIMPLE GROUPS. A.E. Zalesski
|
|
- Frederick Stephens
- 5 years ago
- Views:
Transcription
1 ON LOWER BOUNDS FOR THE DIMENSIONS OF PROJECTIVE MODULES FOR FINITE SIMPLE GROUPS A.E. Zalesski Group rings and Young-Baxter equations Spa, Belgium, June
2 Introduction Let G be a nite group and p a prime. Let F be an algebraically closed eld of char p > 0. Projective indecomposable F G-modules (PIM) are exactly indecomposable direct summands of the regular F G-module. These were introduced by Brauer and Nesbitt in 1940 and remain important objects of study in representation theory of nite groups. However, there are very poor information on their dimensions; an extremal version of the problem can be stated as follows: Problem 1. Given a nite group G and a prime p, determine the minimum dimension of a projective F G-module.
3 The absolute lower bound for the dimension of a PIM is G p, the p-part of the order of G. However, it could be wrong to expect that this bound attains for every group G. There are two well known cases where this is true: 1) G has a subgroup of index G p, in particular, G is p-solvable and 2) G is a Chevalley group (or more generally, a nite reductive group) in dening characteristic p. In fact, the majority of works in the area concern with the decomposition of the characters of projective modules in terms of ordinary irreducible characters, or with computation of decomposition numbers. The matter is that every PIM corresponds to a projective indecomposable K p G-module, where K p is the ring of integers of a suitable nite extension of Q p, the p-adic number eld. The character of this can be expressed in terms of irreducible characters of G. The coecients of this are called the decomposition numbers.
4 In particular, the dimension of a PIM can be expressed in terms of irreducible character degrees of G and the decomposition numbers. This hints that Problem 1 can be studied in the framework of ordinary character theory. The diculty is that no way is known to characterise PIMs in term of ordinary characters. However, as the character of a projective K p G- module vanishes at the p-singular elements, one can somehow ignore this diculty as follows. Denition. An ordinary (reducible) character is called quasi-projective (QP) if it vanishes at the p-singular elements. With this, one can replace Problem 1 by the following one which belongs to the ordinary character theory and is expected to be easier: Problem 2. Determine the minimum degree a QP character of G.
5 Quasi-projective characters A QP-character is called indecomposable if this is not of a sum of proper QP-characters. For groups with cyclic Sylow p-subgroups the indecomposable QP characters are classied in Willems-Z (J. Alg. 2015). In particular, Theorem 1. Let χ be an indecomposable QP character. Then χ = τ + σ, where τ is irreducible, and σ is either 0, or irreducible or the sum of all exceptional characters. In addition, every PIM character is indecomposable as a QP-character. In addition, the indecomposable QP characters have a nice description in terms of the Brauer tree.
6 Obviously, the restriction of a quasi-projective character χ to a Sylow p-subgroup of G is a multilple of the regular character, so χ(1) = l G p, where G p is the p-part of the order of G. We call l the level of χ. This is meaningful for projective characters as well. So G p is the absolute lower bound for the degree of a QP character. The natural problem is: Problem 3. Determine reducible quasi-projective (and projective) characters of level 1. This problem was addressed in a paper by Pellegrini-Z (2016), where it was solved for simple groups of Lie type G for p to be dening characteristic of G, except for types B 3 and D 4. These two exceptional cases are still open.
7 For simple groups Problem 3 was studied by Malle-Weigel (2008), Z. (2013), Pellegrini-Z (2016) and Malle-Z (in preparation). Technics developed there can be used for further progress. In addition, it becomes more clear what kind of diculties arise when dealing with QP characters in comparison with projective ones. To illustrate: Problem 4. Let G = G 1 G 2 be a direct product. Describe indecomposable QP characters in terms of those for G 1, G 2. At least, one cannot expect a simple answer available for projective indecomposable characters, which are the products of those for G 1 and G 2.
8 A less ambitious problem useful for some application is the following: Problem 5. Set G (n) = G G (n times). Suppose that G has no indecomposable QP character of level 1. Is it true that for every m > 0 there exists n such that the levels of all QP characters of G (n) exceed m? For some groups G this is true, and used to prove the following result (Malle-Z, in preparation).
9 Theorem 2. Let p > 2 be a prime and A n the alternating group. Then the minimum degree of a QP character of A n tends to the innity as n. This also implies a similar result for classical groups with p to be a cross characteristic, where the degree in question grows together with the rank n of the group. In constrast, this is not true for the natural characteristic of a classical group. Theorem 2 is shown to be failed for p = 2, however, the analogue of it for classical groups is valid for p = 2. Almost nothing is known for non-simple groups, except for p-solvable.
10 I mention the following result on groups of Lie type (Z, J. Alg. 2013): Theorem 3. Let G = P SL n (p m ), n > 4, and χ be a reducible projective character for the prime p. Then χ(1) (n 1) G p. This bound is sharp. The existence of a projective module of dimension n G p is well known, and for q = 2 there exists such a module of dimension (n 1) G 2. In addition, there exists a QP character of the above degree. I expect that this is projective. Theorem 3 is not probably true for QP characters. In the above paper a similar result is proved also for groups E n (p m ), n = 6, 7, 8, where the bound is shown to be n G p.
11 Malle-Z (in preparation) complete classication of QP characters of degree G p for G simple, except for p = 2 and G = A n. The list is too large to be exposed here. Some bibliography G. Malle and Th. Weigel, Finite groups with minimal 1-PIM, Manuscripta Math. 126 (2008), G. Malle and A. Zalesski, In preparation M. Pellegrini and A. Zalesski, On characters of Chevalley groups vanishing at the non-semisimple elements, Intern. J. Algebra and Comput. Math. 26(2016), M. Pellegrini and A. Zalesski, Irreducible characters of Chevalley groups constant on non-identity unipotent elements, Rend. Sem. Mat. Univ. Padova, Vol. 136(2016), W. Willems and A. Zalesski, Quasi-projective and quasiliftable characters, J. Algebra 442(2015), A. Zalesski, Low dimensional projective indecomposable modules for Chevalley groups in dening characteristic, J. Algebra 377(2013), A. Zalesski, Invariants of maximal tori and unipotent constituents of some quasi-projective characters for nite classical groups, ArXiv: v1 [math.gr] 19 May 2017
`-modular Representations of Finite Reductive Groups
`-modular Representations of Finite Reductive Groups Bhama Srinivasan University of Illinois at Chicago AIM, June 2007 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations AIM,
More informationModular representations of symmetric groups: An Overview
Modular representations of symmetric groups: An Overview Bhama Srinivasan University of Illinois at Chicago Regina, May 2012 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations
More informationEndomorphism rings of permutation modules
Endomorphism rings of permutation modules Natalie Naehrig Lehrstuhl D für Mathematik RWTH Aachen University naehrig@math.rwth-aachen.de December 9, 2009 Abstract Let k be an algebraically closed field
More informationA NOTE ON SPLITTING FIELDS OF REPRESENTATIONS OF FINITE
A NOTE ON SPLITTING FIELDS OF REPRESENTATIONS OF FINITE GROUPS BY Let @ be a finite group, and let x be the character of an absolutely irreducible representation of @ An algebraic number field K is defined
More informationarxiv: v3 [math.gr] 12 Nov 2014
IRREDUCIBLE CHARACTERS OF FINITE SIMPLE GROUPS CONSTANT AT THE p-singular ELEMENTS arxiv:1406.061v3 [math.gr] 1 Nov 014 M.A. PELLEGRINI AND A. ZALESSKI Abstract. In representation theory of finite groups
More informationHeights of characters and defect groups
[Page 1] Heights of characters and defect groups Alexander Moretó 1. Introduction An important result in ordinary character theory is the Ito-Michler theorem, which asserts that a prime p does not divide
More informationON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS
ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,
More informationarxiv: v1 [math.rt] 11 Sep 2009
FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of
More informationTHE 2-MODULAR DECOMPOSITION MATRICES OF THE SYMMETRIC GROUPS S 15, S 16, AND S 17
THE 2-MODULAR DECOMPOSITION MATRICES OF THE SYMMETRIC GROUPS S 15, S 16, AND S 17 Abstract. In this paper the 2-modular decomposition matrices of the symmetric groups S 15, S 16, and S 17 are determined
More information0 A. ... A j GL nj (F q ), 1 j r
CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected
More informationDecomposition Matrix of GL(n,q) and the Heisenberg algebra
Decomposition Matrix of GL(n,q) and the Heisenberg algebra Bhama Srinivasan University of Illinois at Chicago mca-13, August 2013 Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra
More informationarxiv: v1 [math.rt] 14 Nov 2007
arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof
More informationModular Representation Theory of Endomorphism Rings
Modular Representation Theory of Endomorphism Rings Lehrstuhl D für Mathematik RWTH Aachen University Bad Honnef, April 2010 Alperin s Weight Conjecture Assumptions: Let G be a finite group and k be an
More informationModular representation theory
Modular representation theory 1 Denitions for the study group Denition 1.1. Let A be a ring and let F A be the category of all left A-modules. The Grothendieck group of F A is the abelian group dened by
More informationCharacters and triangle generation of the simple Mathieu group M 11
SEMESTER PROJECT Characters and triangle generation of the simple Mathieu group M 11 Under the supervision of Prof. Donna Testerman Dr. Claude Marion Student: Mikaël Cavallin September 11, 2010 Contents
More informationDegree Graphs of Simple Orthogonal and Symplectic Groups
Degree Graphs of Simple Orthogonal and Symplectic Groups Donald L. White Department of Mathematical Sciences Kent State University Kent, Ohio 44242 E-mail: white@math.kent.edu Version: July 12, 2005 In
More informationDUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE
DUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE C. RYAN VINROOT Abstract. We prove that the duality operator preserves the Frobenius- Schur indicators of characters
More informationA Crash Course in Central Simple Algebras
A Crash Course in Central Simple Algebras Evan October 24, 2011 1 Goals This is a prep talk for Danny Neftin's talk. I aim to cover roughly the following topics: (i) Standard results about central simple
More informationHigher dimensional dynamical Mordell-Lang problems
Higher dimensional dynamical Mordell-Lang problems Thomas Scanlon 1 UC Berkeley 27 June 2013 1 Joint with Yu Yasufuku Thomas Scanlon (UC Berkeley) Higher rank DML 27 June 2013 1 / 22 Dynamical Mordell-Lang
More informationLANDAU S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS arxiv: v1 [math.gr] 26 Jun 2015
LANDAU S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS arxiv:1506.08169v1 [math.gr] 26 Jun 2015 MARK L. LEWIS Abstract. When G is solvable group, we prove that the number of conjugacy classes
More informationNAVARRO VERTICES AND NORMAL SUBGROUPS IN GROUPS OF ODD ORDER
NAVARRO VERTICES AND NORMAL SUBGROUPS IN GROUPS OF ODD ORDER JAMES P. COSSEY Abstract. Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro
More informationAnisotropic Groups over Arbitrary Fields* Ulf Rehmann Why anisotropic groups? 1888/9 Killing classifies semisimple groups and introduces the types A
* Ulf Rehmann Why anisotropic groups? 1888/9 Killing classifies semisimple groups and introduces the types A n, B n, C n, D n, E 6, E 7, E 8, F 4, G 2 of semisimple Lie groups. 1961 Chevalley shows: These
More informationLifting to non-integral idempotents
Journal of Pure and Applied Algebra 162 (2001) 359 366 www.elsevier.com/locate/jpaa Lifting to non-integral idempotents Georey R. Robinson School of Mathematics and Statistics, University of Birmingham,
More informationSELF-EQUIVALENCES OF THE DERIVED CATEGORY OF BRAUER TREE ALGEBRAS WITH EXCEPTIONAL VERTEX
An. Şt. Univ. Ovidius Constanţa Vol. 9(1), 2001, 139 148 SELF-EQUIVALENCES OF THE DERIVED CATEGORY OF BRAUER TREE ALGEBRAS WITH EXCEPTIONAL VERTEX Alexander Zimmermann Abstract Let k be a field and A be
More informationOn a question of B.H. Neumann
On a question of B.H. Neumann Robert Guralnick Department of Mathematics University of Southern California E-mail: guralnic@math.usc.edu Igor Pak Department of Mathematics Massachusetts Institute of Technology
More informationUniversität Kassel, Universitat de València, University of Copenhagen
ZEROS OF CHARACTERS OF FINITE GROUPS Gunter Malle, Gabriel Navarro and Jørn B. Olsson Universität Kassel, Universitat de València, University of Copenhagen 1. Introduction Let G be a finite group and let
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationConjugacy Classes in Semisimple Algebraic Groups (Revisions)
Conjugacy Classes in Semisimple Algebraic Groups (Revisions) Subtract 4 from each page number in the Index. (This production error seems to be corrected in the paperback reprint.) 7 In line 3 replace M
More information294 Meinolf Geck In 1992, Lusztig [16] addressed this problem in the framework of his theory of character sheaves and its application to Kawanaka's th
Doc. Math. J. DMV 293 On the Average Values of the Irreducible Characters of Finite Groups of Lie Type on Geometric Unipotent Classes Meinolf Geck Received: August 16, 1996 Communicated by Wolfgang Soergel
More informationON BRAUER'S HEIGHT 0 CONJECTURE
T. R. Berger and R. Knδrr Nag*oya Math. J. Vol. 109 (1988), 109-116 ON BRAUER'S HEIGHT 0 CONJECTURE T.R. BERGER AND R. KNϋRR R. Brauer not only laid the foundations of modular representation theory of
More informationCONJECTURES ON CHARACTER DEGREES FOR THE SIMPLE THOMPSON GROUP
Uno, K. Osaka J. Math. 41 (2004), 11 36 CONJECTURES ON CHARACTER DEGREES FOR THE SIMPLE THOMPSON GROUP Dedicated to Professor Yukio Tsushima on his sixtieth birthday KATSUHIRO UNO 1. Introduction (Received
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationPrimitive Ideals and Unitarity
Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)
More informationarxiv:math/ v3 [math.qa] 16 Feb 2003
arxiv:math/0108176v3 [math.qa] 16 Feb 2003 UNO S CONJECTURE ON REPRESENTATION TYPES OF HECKE ALGEBRAS SUSUMU ARIKI Abstract. Based on a recent result of the author and A.Mathas, we prove that Uno s conjecture
More informationFundamental Lemma and Hitchin Fibration
Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental
More informationTHE 7-MODULAR DECOMPOSITION MATRICES OF THE SPORADIC O NAN GROUP
THE 7-MODULAR DECOMPOSITION MATRICES OF THE SPORADIC O NAN GROUP ANNE HENKE, GERHARD HISS, AND JÜRGEN MÜLLER Abstract. The determination of the modular character tables of the sporadic O Nan group, its
More informationCENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS
Actes, Congrès intern, math., 1970. Tome 1, p. 355 à 360. CENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS by DANIEL GORENSTEIN It has been known for a long time that the structure of a finite simple
More informationALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS
ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample
More informationQUALIFYING EXAM IN ALGEBRA August 2011
QUALIFYING EXAM IN ALGEBRA August 2011 1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra 1 problem II. Group Theory 3 problems III. Ring
More informationClassifying Camina groups: A theorem of Dark and Scoppola
Classifying Camina groups: A theorem of Dark and Scoppola arxiv:0807.0167v5 [math.gr] 28 Sep 2011 Mark L. Lewis Department of Mathematical Sciences, Kent State University Kent, Ohio 44242 E-mail: lewis@math.kent.edu
More informationGroup Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.
Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite
More informationON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS
Submitted exclusively to the London Mathematical Society DOI: 0./S0000000000000000 ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS JOHN N. BRAY and ROBERT A. WILSON Abstract In the Kourovka Notebook,
More informationSIMPLE GROUPS ADMIT BEAUVILLE STRUCTURES
SIMPLE GROUPS ADMIT BEAUVILLE STRUCTURES ROBERT GURALNICK AND GUNTER MALLE Dedicated to the memory of Fritz Grunewald Abstract. We prove a conjecture of Bauer, Catanese and Grunewald showing that all finite
More informationMaximal subgroups of exceptional groups: representing groups in groups
Maximal subgroups of exceptional groups: representing groups in groups David A. Craven University of Birmingham Representations of Finite and Algebraic Groups, Les Houches. 9th February, 2015. Joint with
More informationThe Major Problems in Group Representation Theory
The Major Problems in Group Representation Theory David A. Craven 18th November 2009 In group representation theory, there are many unsolved conjectures, most of which try to understand the involved relationship
More informationR E N D I C O N T I PRIME GRAPH COMPONENTS OF FINITE ALMOST SIMPLE GROUPS
R E N D I C O N T I del Seminario Matematico dell Università di Padova Vol. 102 Anno 1999 PRIME GRAPH COMPONENTS OF FINITE ALMOST SIMPLE GROUPS Maria Silvia Lucido Dipartimento di Matematica Pura e Applicata
More informationGroup Gradings on Finite Dimensional Lie Algebras
Algebra Colloquium 20 : 4 (2013) 573 578 Algebra Colloquium c 2013 AMSS CAS & SUZHOU UNIV Group Gradings on Finite Dimensional Lie Algebras Dušan Pagon Faculty of Natural Sciences and Mathematics, University
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/22043 holds various files of this Leiden University dissertation. Author: Anni, Samuele Title: Images of Galois representations Issue Date: 2013-10-24 Chapter
More informationCentralizers and the maximum size of the pairwise noncommuting elements in nite groups
Hacettepe Journal of Mathematics and Statistics Volume 46 (2) (2017), 193 198 Centralizers and the maximum size of the pairwise noncommuting elements in nite groups Seyyed Majid Jafarian Amiri and Hojjat
More informationON THE CHIEF FACTORS OF PARABOLIC MAXIMAL SUBGROUPS IN FINITE SIMPLE GROUPS OF NORMAL LIE TYPE
Siberian Mathematical Journal, Vol. 55, No. 4, pp. 622 638, 2014 Original Russian Text Copyright c 2014 Korableva V.V. ON THE CHIEF FACTORS OF PARABOLIC MAXIMAL SUBGROUPS IN FINITE SIMPLE GROUPS OF NORMAL
More informationCentralizers of Finite Subgroups in Simple Locally Finite Groups
Centralizers of Finite Subgroups in Simple Locally Finite Groups Kıvanc. Ersoy 1 Simple Locally Finite Groups A group G is called locally finite if every finitely generated subgroup of G is finite. In
More informationAbsolutely indecomposable symmetric matrices
Journal of Pure and Applied Algebra 174 (2002) 83 93 wwwelseviercom/locate/jpaa Absolutely indecomposable symmetric matrices Hans A Keller a; ;1, A Herminia Ochsenius b;1 a Hochschule Technik+Architektur
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More informationINDECOMPOSABLE LIE ALGEBRAS WITH NONTRIVIAL LEVI DECOMPOSITION CANNOT HAVE FILIFORM RADICAL
International Mathematical Forum, 1, 2006, no. 7, 309-316 INDECOMPOSABLE LIE ALGEBRAS WITH NONTRIVIAL LEVI DECOMPOSITION CANNOT HAVE FILIFORM RADICAL J. M. Ancochea Bermúdez Dpto. Geometría y Topología
More informationContents. Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39.
Preface xiii Chapter 1. Selected Problems in One Complex Variable 1 1.1. Preliminaries 2 1.2. A Simple Problem 2 1.3. Partitions of Unity 4 1.4. The Cauchy-Riemann Equations 7 1.5. The Proof of Proposition
More informationAlgebras and regular subgroups. Marco Antonio Pellegrini. Ischia Group Theory Joint work with Chiara Tamburini
Joint work with Chiara Tamburini Università Cattolica del Sacro Cuore Ischia Group Theory 2016 Affine group and Regular subgroups Let F be any eld. We identify the ane group AGL n (F) with the subgroup
More informationA NOTE ON RETRACTS AND LATTICES (AFTER D. J. SALTMAN)
A NOTE ON RETRACTS AND LATTICES (AFTER D. J. SALTMAN) Z. REICHSTEIN Abstract. This is an expository note based on the work of D. J. Saltman. We discuss the notions of retract rationality and retract equivalence
More informationSylow structure of finite groups
Sylow structure of finite groups Jack Schmidt University of Kentucky September 2, 2009 Abstract: Probably the most powerful results in the theory of finite groups are the Sylow theorems. Those who have
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationImplications of the index of a fixed point subgroup
Rend. Sem. Mat. Univ. Padova, DRAFT, 1 7 Implications of the index of a fixed point subgroup Erkan Murat Türkan ( ) Abstract Let G be a finite group and A Aut(G). The index G : C G (A) is called the index
More informationTHREE CASES AN EXAMPLE: THE ALTERNATING GROUP A 5
THREE CASES REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE LECTURE II: DELIGNE-LUSZTIG THEORY AND SOME APPLICATIONS Gerhard Hiss Lehrstuhl D für Mathematik RWTH Aachen University Summer School Finite Simple
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationA MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 7, 2017
A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET DIPENDRA PRASAD March 7, 2017 Abstract. Following the natural instinct that when a group operates on a number field then every term in the
More informationNew York Journal of Mathematics New York J. Math. 5 (1999) 115{120. Explicit Local Heights Graham Everest Abstract. A new proof is given for the expli
New York Journal of Mathematics New York J. Math. 5 (1999) 115{10. Explicit Local eights raham Everest Abstract. A new proof is given for the explicit formulae for the non-archimedean canonical height
More informationALGEBRA 11: Galois theory
Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic
More informationOn p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree. Christine Bessenrodt
On p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree Christine Bessenrodt Institut für Algebra, Zahlentheorie und Diskrete Mathematik Leibniz Universität
More informationEQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms
EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be
More informationSpherical varieties and arc spaces
Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January 2017 1 Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected
More informationCategory O and its basic properties
Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α
More informationThe Morozov-Jacobson Theorem on 3-dimensional Simple Lie Subalgebras
The Morozov-Jacobson Theorem on 3-dimensional Simple Lie Subalgebras Klaus Pommerening July 1979 english version April 2012 The Morozov-Jacobson theorem says that every nilpotent element of a semisimple
More informationIrreducible subgroups of algebraic groups
Irreducible subgroups of algebraic groups Martin W. Liebeck Department of Mathematics Imperial College London SW7 2BZ England Donna M. Testerman Department of Mathematics University of Lausanne Switzerland
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationWeyl group representations on zero weight spaces
Weyl group representations on zero weight spaces November 30, 2014 Here we survey briefly (trying to provide reasonably complete references) the scattered work over four decades most relevant to the indicated
More informationON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig
ON SOME PARTITIONS OF A FLAG MANIFOLD G. Lusztig Introduction Let G be a connected reductive group over an algebraically closed field k of characteristic p 0. Let W be the Weyl group of G. Let W be the
More informationFIXED POINT SETS AND LEFSCHETZ MODULES. John Maginnis and Silvia Onofrei Department of Mathematics, Kansas State University
AMS Sectional Meeting Middle Tennessee State University, Murfreesboro, TN 3-4 November 2007 FIXED POINT SETS AND LEFSCHETZ MODULES John Maginnis and Silvia Onofrei Department of Mathematics, Kansas State
More informationLecture 20 FUNDAMENTAL Theorem of Finitely Generated Abelian Groups (FTFGAG)
Lecture 20 FUNDAMENTAL Theorem of Finitely Generated Abelian Groups (FTFGAG) Warm up: 1. Let n 1500. Find all sequences n 1 n 2... n s 2 satisfying n i 1 and n 1 n s n (where s can vary from sequence to
More informationCharacter values and decomposition matrices of symmetric groups
Character values and decomposition matrices of symmetric groups Mark Wildon Abstract The relationships between the values taken by ordinary characters of symmetric groups are exploited to prove two theorems
More informationMath 249B. Nilpotence of connected solvable groups
Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C
More informationRECOLLECTION THE BRAUER CHARACTER TABLE
RECOLLECTION REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE LECTURE III: REPRESENTATIONS IN NON-DEFINING CHARACTERISTICS AIM Classify all irreducible representations of all finite simple groups and related
More informationAlgebra Exam Topics. Updated August 2017
Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have
More informationFOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2
FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued
More informationNotes on D 4 May 7, 2009
Notes on D 4 May 7, 2009 Consider the simple Lie algebra g of type D 4 over an algebraically closed field K of characteristic p > h = 6 (the Coxeter number). In particular, p is a good prime. We have dim
More information14 From modular forms to automorphic representations
14 From modular forms to automorphic representations We fix an even integer k and N > 0 as before. Let f M k (N) be a modular form. We would like to product a function on GL 2 (A Q ) out of it. Recall
More informationSTATE OF THE ART OF THE OPEN PROBLEMS IN: MODULE THEORY. ENDOMORPHISM RINGS AND DIRECT SUM DECOMPOSITIONS IN SOME CLASSES OF MODULES
STATE OF THE ART OF THE OPEN PROBLEMS IN: MODULE THEORY. ENDOMORPHISM RINGS AND DIRECT SUM DECOMPOSITIONS IN SOME CLASSES OF MODULES ALBERTO FACCHINI In Chapter 11 of my book Module Theory. Endomorphism
More informationHEIGHT 0 CHARACTERS OF FINITE GROUPS OF LIE TYPE
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 00, Pages 000 000 (Xxxx XX, XXXX) S 1088-4165(XX)0000-0 HEIGHT 0 CHARACTERS OF FINITE GROUPS OF LIE TYPE GUNTER MALLE
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationThe Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete
The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete Kyle Riggs Indiana University April 29, 2014 Kyle Riggs Decomposability Problem 1/ 22 Computable Groups background A group
More informationON THE PLESKEN LIE ALGEBRA DEFINED OVER A FINITE FIELD
ON THE PLESKEN LIE ALGEBRA DEFINED OVER A FINITE FIELD JOHN CULLINAN AND MONA MERLING Abstract Let G be a finite group and p a prime number The Plesken Lie algebra is a subalgebra of the complex group
More informationCharacter Theory and Group Rings
Contemporary Mathematics Character Theory and Group Rings D. S. Passman Dedicated to Professor I. Martin Isaacs Abstract. While we were graduate students, Marty Isaacs and I worked together on the character
More informationSubgroups of Linear Algebraic Groups
Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples 5 1.1. Introduction to Linear Algebraic Groups 5 1.2.
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationAlgebra Questions. May 13, Groups 1. 2 Classification of Finite Groups 4. 3 Fields and Galois Theory 5. 4 Normal Forms 9
Algebra Questions May 13, 2013 Contents 1 Groups 1 2 Classification of Finite Groups 4 3 Fields and Galois Theory 5 4 Normal Forms 9 5 Matrices and Linear Algebra 10 6 Rings 11 7 Modules 13 8 Representation
More informationA GENERALISATION OF A THEOREM OF WIELANDT
A GENERALISATION OF A THEOREM OF WIELANDT FRANCESCO FUMAGALLI AND GUNTER MALLE Abstract. In 1974, Helmut Wielandt proved that in a nite group G, a subgroup A is subnormal if and only if it is subnormal
More informationON AN INFINITE-DIMENSIONAL GROUP OVER A FINITE FIELD. A. M. Vershik, S. V. Kerov
ON AN INFINITE-DIMENSIONAL GROUP OVER A FINITE FIELD A. M. Vershik, S. V. Kerov Introduction. The asymptotic representation theory studies the behavior of representations of large classical groups and
More informationarxiv: v2 [math.co] 6 Oct 2016
ON THE CRITICAL GROUP OF THE MISSING MOORE GRAPH. arxiv:1509.00327v2 [math.co] 6 Oct 2016 JOSHUA E. DUCEY Abstract. We consider the critical group of a hypothetical Moore graph of diameter 2 and valency
More informationProof. We omit the proof of the case when f is the reduction of a characteristic zero eigenform (cf. Theorem 4.1 in [DS74]), so assume P (X) has disti
Some local (at p) properties of residual Galois representations Johnson Jia, Krzysztof Klosin March 5, 26 1 Preliminary results In this talk we are going to discuss some local properties of (mod p) Galois
More informationTHEORY OF GROUP REPRESENTATIONS AND APPLICATIONS
THEORY OF GROUP REPRESENTATIONS AND APPLICATIONS ASIM 0. BARUT Institute for Theoretical Physics, University of Colorado, Boulder, Colo., U.S.A. RYSZARD RATJZKA Institute for Nuclear Research, Warszawa,
More informationON THE DISTRIBUTION OF CLASS GROUPS OF NUMBER FIELDS
ON THE DISTRIBUTION OF CLASS GROUPS OF NUMBER FIELDS GUNTER MALLE Abstract. We propose a modification of the predictions of the Cohen Lenstra heuristic for class groups of number fields in the case where
More informationFields of cohomological dimension 1 versus C 1 -fields
Fields of cohomological dimension 1 versus C 1 -fields J.-L. Colliot-Thélène Abstract. Ax gave examples of fields of cohomological dimension 1 which are not C 1 -fields. Kato and Kuzumaki asked whether
More informationMATH 527, Topics in Representation Theory
Archive of documentation for MATH 527, Topics in Representation Theory Bilkent University, Spring 2016, Laurence Barker version: 20 May 2016 Source file: arch527spr16.tex page 2: Course specification.
More information