Representations of Algebraic Groups
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ORTHOGONAL GROUP of CERTAIN INDEFINITE LATTICE Chang Heon Kim* 1. Introduction Given an Even Lattice M in a Real Quadratic Space
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No. 1, March 2007 ORTHOGONAL GROUP OF CERTAIN INDEFINITE LATTICE Chang Heon Kim* Abstract. We compute the special orthogonal group of certain lattice of signature (2; 1). 1. Introduction Given an even lattice M in a real quadratic space of signature (2; n), Borcherds lifting [1] gives a multiplicative correspondence between vec- tor valued modular forms F of weight 1¡n=2 with values in C[M 0=M] (= the group ring of M 0=M) and meromorphic modular forms on complex 0 varieties (O(2) £ O(n))nO(2; n)=Aut(M; F ). Here NM denotes the dual lattice of M, O(2; n) is the orthogonal group of M R and Aut(M; F ) is the subgroup of Aut(M) leaving the form F stable under the natural action of Aut(M) on M 0=M. In particular, if the signature of M is (2; 1), then O(2; 1) ¼ H: O(2) £ O(1) and Borcherds' theory gives a lifting of vector valued modular form of weight 1=2 to usual one variable modular form on Aut(M; F ). In this sense in order to work out Borcherds lifting it is important to ¯nd appropriate lattice on which our wanted modular group acts. In this article we will show: Theorem 1.1. Let M be a 3-dimensional even lattice of all 2 £ 2 integral symmetric matrices, that is, ½µ ¶ ¾ AB M = j A; B; C 2 Z BC Received December 30, 2006. 2000 Mathematics Subject Classi¯cation: Primary 11F03, 11H56. -
ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆. -
REPRESENTATIONS of FINITE GROUPS 1. Definition And
REPRESENTATIONS OF FINITE GROUPS 1. Definition and Introduction 1.1. Definitions. Let V be a vector space over the field C of complex numbers and let GL(V ) be the group of isomorphisms of V onto itself. An element a of GL(V ) is a linear mapping of V into V which has a linear inverse a−1. When V has dimension n and has n a finite basis (ei)i=1, each map a : V ! V is defined by a square matrix (aij) of order n. The coefficients aij are complex numbers. They are obtained by expressing the image a(ej) in terms of the basis (ei): X a(ej) = aijei: i Remark 1.1. Saying that a is an isomorphism is equivalent to saying that the determinant det(a) = det(aij) of a is not zero. The group GL(V ) is thus identified with the group of invertible square matrices of order n. Suppose G is a finite group with identity element 1. A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism ρ : G ! GL(V ) of G to the group of automorphisms of V . We say such a map gives V the structure of a G-module. The dimension V is called the degree of ρ. Remark 1.2. When there is little ambiguity of the map ρ, we sometimes call V itself a representation of G. In this vein we often write gv_ or gv for ρ(g)v. Remark 1.3. Since ρ is a homomorphism, we have equality ρ(st) = ρ(s)ρ(t) for s; t 2 G: In particular, we have ρ(1) = 1; ; ρ(s−1) = ρ(s)−1: A map ' between two representations V and W of G is a vector space map ' : V ! W such that ' V −−−−! W ? ? g? ?g (1.1) y y V −−−−! W ' commutes for all g 2 G. -
Notes on the Finite Fourier Transform
Notes on the Finite Fourier Transform Peter Woit Department of Mathematics, Columbia University [email protected] April 28, 2020 1 Introduction In this final section of the course we'll discuss a topic which is in some sense much simpler than the cases of Fourier series for functions on S1 and the Fourier transform for functions on R, since it will involve functions on a finite set, and thus just algebra and no need for the complexities of analysis. This topic has important applications in the approximate computation of Fourier series (which we won't cover), and in number theory (which we'll say a little bit about). 2 The group Z(N) What we'll be doing is simplifying the topic of Fourier series by replacing the multiplicative group S1 = fz 2 C : jzj = 1g by the finite subgroup Z(N) = fz 2 C : zN = 1g If we write z = reiθ, then zN = 1 =) rN eiθN = 1 =) r = 1; θN = k2π (k 2 Z) k =) θ = 2π k = 0; 1; 2; ··· ;N − 1 N mod 2π The group Z(N) is thus explicitly given by the set i 2π i2 2π i(N−1) 2π Z(N) = f1; e N ; e N ; ··· ; e N g Geometrically, these are the points on the unit circle one gets by starting at 1 2π and dividing it into N sectors with equal angles N . (Draw a picture, N = 6) The set Z(N) is a group, with 1 identity element 1 (k = 0) inverse 2πi k −1 −2πi k 2πi (N−k) (e N ) = e N = e N multiplication law ik 2π il 2π i(k+l) 2π e N e N = e N One can equally well write this group as the additive group (Z=NZ; +) of inte- gers mod N, with isomorphism Z(N) $ Z=NZ ik 2π e N $ [k]N 1 $ [0]N −ik 2π e N $ −[k]N = [−k]N = [N − k]N 3 Fourier analysis on Z(N) An abstract point of view on the theory of Fourier series is that it is based on exploiting the existence of a particular othonormal basis of functions on the group S1 that are eigenfunctions of the linear transformations given by rotations. -
A NEW APPROACH to RANK ONE LINEAR ALGEBRAIC GROUPS An
A NEW APPROACH TO RANK ONE LINEAR ALGEBRAIC GROUPS DANIEL ALLCOCK Abstract. One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material before one can even characterize PGL2. Our goal here is to show how to develop the root system, etc., using only the completeness of the flag variety, its immediate consequences, and some facts about solvable groups. In particular, one can skip over the usual analysis of Cartan subgroups, the fact that G is the union of its Borel subgroups, the connectedness of torus centralizers, and the normalizer theorem (that Borel subgroups are self-normalizing). The main idea is a new approach to the structure of rank 1 groups; the key step is lemma 5. All algebraic geometry is over a fixed algebraically closed field. G always denotes a connected linear algebraic group with Lie algebra g, T a maximal torus, and B a Borel subgroup containing it. We assume the structure theory for connected solvable groups, and the completeness of the flag variety G/B and some of its consequences. Namely: that all Borel subgroups (resp. maximal tori) are conjugate; that G is nilpotent if one of its Borel subgroups is; that CG(T )0 lies in every Borel subgroup containing T ; and that NG(B) contains B of finite index and (therefore) is self-normalizing. -
An Introduction to the Trace Formula
Clay Mathematics Proceedings Volume 4, 2005 An Introduction to the Trace Formula James Arthur Contents Foreword 3 Part I. The Unrefined Trace Formula 7 1. The Selberg trace formula for compact quotient 7 2. Algebraic groups and adeles 11 3. Simple examples 15 4. Noncompact quotient and parabolic subgroups 20 5. Roots and weights 24 6. Statement and discussion of a theorem 29 7. Eisenstein series 31 8. On the proof of the theorem 37 9. Qualitative behaviour of J T (f) 46 10. The coarse geometric expansion 53 11. Weighted orbital integrals 56 12. Cuspidal automorphic data 64 13. A truncation operator 68 14. The coarse spectral expansion 74 15. Weighted characters 81 Part II. Refinements and Applications 89 16. The first problem of refinement 89 17. (G, M)-families 93 18. Localbehaviourofweightedorbitalintegrals 102 19. The fine geometric expansion 109 20. Application of a Paley-Wiener theorem 116 21. The fine spectral expansion 126 22. The problem of invariance 139 23. The invariant trace formula 145 24. AclosedformulaforthetracesofHeckeoperators 157 25. Inner forms of GL(n) 166 Supported in part by NSERC Discovery Grant A3483. c 2005 Clay Mathematics Institute 1 2 JAMES ARTHUR 26. Functoriality and base change for GL(n) 180 27. The problem of stability 192 28. Localspectraltransferandnormalization 204 29. The stable trace formula 216 30. Representationsofclassicalgroups 234 Afterword: beyond endoscopy 251 References 258 Foreword These notes are an attempt to provide an entry into a subject that has not been very accessible. The problems of exposition are twofold. It is important to present motivation and background for the kind of problems that the trace formula is designed to solve. -
New Publications Offered by the AMS
New Publications Offered by the AMS appropriate generality waited for the seventies. These early Algebra and Algebraic occurrences were in algebraic topology in the study of (iter- ated) loop spaces and their chain algebras. In the nineties, Geometry there was a renaissance and further development of the theory inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, Almost Commuting derived categories, Morse theory, symplectic and contact EMOIRS M of the geometry, combinatorics, knot theory, moduli spaces, cyclic American Mathematical Society Elements in Compact cohomology, and, not least, theoretical physics, especially Volume 157 Number 747 string field theory and deformation quantization. The general- Almost Commuting Lie Groups Elements in ization of quadratic duality (e.g., Lie algebras as dual to Compact Lie Groups Armand Borel, Institute for commutative algebras) together with the property of Koszul- Armand Borel Advanced Study, Princeton, NJ, ness in an essentially operadic context provided an additional Robert Friedman computational tool for studying homotopy properties outside John W. Morgan and Robert Friedman and THEMAT A IC M A L N A S O C I C of the topological setting. R I E E T M Y A FO 8 U 88 John W. Morgan, Columbia NDED 1 University, New York City, NY The book contains a detailed and comprehensive historical American Mathematical Society introduction describing the development of operad theory Contents: Introduction; Almost from the initial period when it was a rather specialized tool in commuting N-tuples; Some characterizations of groups of homotopy theory to the present when operads have a wide type A; c-pairs; Commuting triples; Some results on diagram range of applications in algebra, topology, and mathematical automorphisms and associated root systems; The fixed physics. -
THE UPPER BOUND of COMPOSITION SERIES 1. Introduction. a Composition Series, That Is a Series of Subgroups Each Normal in the Pr
THE UPPER BOUND OF COMPOSITION SERIES ABHIJIT BHATTACHARJEE Abstract. In this paper we prove that among all finite groups of or- α1 α2 αr der n2 N with n ≥ 2 where n = p1 p2 :::pr , the abelian group with elementary abelian sylow subgroups, has the highest number of composi- j Pr α ! Qr Qαi pi −1 ( i=1 i) tion series and it has i=1 j=1 Qr distinct composition pi−1 i=1 αi! series. We also prove that among all finite groups of order ≤ n, n 2 N α with n ≥ 4, the elementary abelian group of order 2 where α = [log2 n] has the highest number of composition series. 1. Introduction. A composition series, that is a series of subgroups each normal in the previous such that corresponding factor groups are simple. Any finite group has a composition series. The famous Jordan-Holder theo- rem proves that, the composition factors in a composition series are unique up to isomorphism. Sometimes a group of small order has a huge number of distinct composition series. For example an elementary abelian group of order 64 has 615195 distinct composition series. In [6] there is an algo- rithm in GAP to find the distinct composition series of any group of finite order.The aim of this paper is to provide an upper bound of the number of distinct composition series of any group of finite order. The approach of this is combinatorial and the method is elementary. All the groups considered in this paper are of finite order. 2. Some Basic Results On Composition Series. -
Weak Representation Theory in the Calculus of Relations Jeremy F
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2006 Weak representation theory in the calculus of relations Jeremy F. Alm Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Alm, Jeremy F., "Weak representation theory in the calculus of relations " (2006). Retrospective Theses and Dissertations. 1795. https://lib.dr.iastate.edu/rtd/1795 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Weak representation theory in the calculus of relations by Jeremy F. Aim A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Roger Maddux, Major Professor Maria Axenovich Paul Sacks Jonathan Smith William Robinson Iowa State University Ames, Iowa 2006 Copyright © Jeremy F. Aim, 2006. All rights reserved. UMI Number: 3217250 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. -
On Some Generation Methods of Finite Simple Groups
Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography On Some Generation Methods of Finite Simple Groups Ayoub B. M. Basheer Department of Mathematical Sciences, North-West University (Mafikeng), P Bag X2046, Mmabatho 2735, South Africa Groups St Andrews 2017 in Birmingham, School of Mathematics, University of Birmingham, United Kingdom 11th of August 2017 Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Abstract In this talk we consider some methods of generating finite simple groups with the focus on ranks of classes, (p; q; r)-generation and spread (exact) of finite simple groups. We show some examples of results that were established by the author and his supervisor, Professor J. Moori on generations of some finite simple groups. Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Introduction Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, Di Martino and et al. [39] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as generators carrying a geometric meaning, generators of some prescribed order, generators that offer an economical presentation of the group. -
Geometric Representation Theory, Fall 2005
GEOMETRIC REPRESENTATION THEORY, FALL 2005 Some references 1) J. -P. Serre, Complex semi-simple Lie algebras. 2) T. Springer, Linear algebraic groups. 3) A course on D-modules by J. Bernstein, availiable at www.math.uchicago.edu/∼arinkin/langlands. 4) J. Dixmier, Enveloping algebras. 1. Basics of the category O 1.1. Refresher on semi-simple Lie algebras. In this course we will work with an alge- braically closed ground field of characteristic 0, which may as well be assumed equal to C Let g be a semi-simple Lie algebra. We will fix a Borel subalgebra b ⊂ g (also sometimes denoted b+) and an opposite Borel subalgebra b−. The intersection b+ ∩ b− is a Cartan subalgebra, denoted h. We will denote by n, and n− the unipotent radicals of b and b−, respectively. We have n = [b, b], and h ' b/n. (I.e., h is better to think of as a quotient of b, rather than a subalgebra.) The eigenvalues of h acting on n are by definition the positive roots of g; this set will be denoted by ∆+. We will denote by Q+ the sub-semigroup of h∗ equal to the positive span of ∆+ (i.e., Q+ is the set of eigenvalues of h under the adjoint action on U(n)). For λ, µ ∈ h∗ we shall say that λ ≥ µ if λ − µ ∈ Q+. We denote by P + the sub-semigroup of dominant integral weights, i.e., those λ that satisfy hλ, αˇi ∈ Z+ for all α ∈ ∆+. + For α ∈ ∆ , we will denote by nα the corresponding eigen-space. -
14.2 Covers of Symmetric and Alternating Groups
Remark 206 In terms of Galois cohomology, there an exact sequence of alge- braic groups (over an algebrically closed field) 1 → GL1 → ΓV → OV → 1 We do not necessarily get an exact sequence when taking values in some subfield. If 1 → A → B → C → 1 is exact, 1 → A(K) → B(K) → C(K) is exact, but the map on the right need not be surjective. Instead what we get is 1 → H0(Gal(K¯ /K), A) → H0(Gal(K¯ /K), B) → H0(Gal(K¯ /K), C) → → H1(Gal(K¯ /K), A) → ··· 1 1 It turns out that H (Gal(K¯ /K), GL1) = 1. However, H (Gal(K¯ /K), ±1) = K×/(K×)2. So from 1 → GL1 → ΓV → OV → 1 we get × 1 1 → K → ΓV (K) → OV (K) → 1 = H (Gal(K¯ /K), GL1) However, taking 1 → µ2 → SpinV → SOV → 1 we get N × × 2 1 ¯ 1 → ±1 → SpinV (K) → SOV (K) −→ K /(K ) = H (K/K, µ2) so the non-surjectivity of N is some kind of higher Galois cohomology. Warning 207 SpinV → SOV is onto as a map of ALGEBRAIC GROUPS, but SpinV (K) → SOV (K) need NOT be onto. Example 208 Take O3(R) =∼ SO3(R)×{±1} as 3 is odd (in general O2n+1(R) =∼ SO2n+1(R) × {±1}). So we have a sequence 1 → ±1 → Spin3(R) → SO3(R) → 1. 0 × Notice that Spin3(R) ⊆ C3 (R) =∼ H, so Spin3(R) ⊆ H , and in fact we saw that it is S3. 14.2 Covers of symmetric and alternating groups The symmetric group on n letter can be embedded in the obvious way in On(R) as permutations of coordinates.