INTEGER POINTS and THEIR ORTHOGONAL LATTICES 2 to Remove the Congruence Condition
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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 ∆. -
7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes. -
Unitary Group - Wikipedia
Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1. -
Groups with Identical Subgroup Lattices in All Powers
GROUPS WITH IDENTICAL SUBGROUP LATTICES IN ALL POWERS KEITH A. KEARNES AND AGNES´ SZENDREI Abstract. Suppose that G and H are groups with cyclic Sylow subgroups. We show that if there is an isomorphism λ2 : Sub (G × G) ! Sub (H × H), then there k k are isomorphisms λk : Sub (G ) ! Sub (H ) for all k. But this is not enough to force G to be isomorphic to H, for we also show that for any positive integer N there are pairwise nonisomorphic groups G1; : : : ; GN defined on the same finite set, k k all with cyclic Sylow subgroups, such that Sub (Gi ) = Sub (Gj ) for all i; j; k. 1. Introduction To what extent is a finite group determined by the subgroup lattices of its finite direct powers? Reinhold Baer proved results in 1939 implying that an abelian group G is determined up to isomorphism by Sub (G3) (cf. [1]). Michio Suzuki proved in 1951 that a finite simple group G is determined up to isomorphism by Sub (G2) (cf. [10]). Roland Schmidt proved in 1981 that if G is a finite, perfect, centerless group, then it is determined up to isomorphism by Sub (G2) (cf. [6]). Later, Schmidt proved in [7] that if G has an elementary abelian Hall normal subgroup that equals its own centralizer, then G is determined up to isomorphism by Sub (G3). It has long been open whether every finite group G is determined up to isomorphism by Sub (G3). (For more information on this problem, see the books [8, 11].) One may ask more generally to what extent a finite algebraic structure (or algebra) is determined by the subalgebra lattices of its finite direct powers. -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
10 Group Theory
10 Group theory 10.1 What is a group? A group G is a set of elements f, g, h, ... and an operation called multipli- cation such that for all elements f,g, and h in the group G: 1. The product fg is in the group G (closure); 2. f(gh)=(fg)h (associativity); 3. there is an identity element e in the group G such that ge = eg = g; 1 1 1 4. every g in G has an inverse g− in G such that gg− = g− g = e. Physical transformations naturally form groups. The elements of a group might be all physical transformations on a given set of objects that leave invariant a chosen property of the set of objects. For instance, the objects might be the points (x, y) in a plane. The chosen property could be their distances x2 + y2 from the origin. The physical transformations that leave unchanged these distances are the rotations about the origin p x cos ✓ sin ✓ x 0 = . (10.1) y sin ✓ cos ✓ y ✓ 0◆ ✓− ◆✓ ◆ These rotations form the special orthogonal group in 2 dimensions, SO(2). More generally, suppose the transformations T,T0,T00,... change a set of objects in ways that leave invariant a chosen property property of the objects. Suppose the product T 0 T of the transformations T and T 0 represents the action of T followed by the action of T 0 on the objects. Since both T and T 0 leave the chosen property unchanged, so will their product T 0 T . Thus the closure condition is satisfied. -
Matrix Lie Groups and Their Lie Algebras
Matrix Lie groups and their Lie algebras Alen Alexanderian∗ Abstract We discuss matrix Lie groups and their corresponding Lie algebras. Some common examples are provided for purpose of illustration. 1 Introduction The goal of these brief note is to provide a quick introduction to matrix Lie groups which are a special class of abstract Lie groups. Study of matrix Lie groups is a fruitful endeavor which allows one an entry to theory of Lie groups without requiring knowl- edge of differential topology. After all, most interesting Lie groups turn out to be matrix groups anyway. An abstract Lie group is defined to be a group which is also a smooth manifold, where the group operations of multiplication and inversion are also smooth. We provide a much simple definition for a matrix Lie group in Section 4. Showing that a matrix Lie group is in fact a Lie group is discussed in standard texts such as [2]. We also discuss Lie algebras [1], and the computation of the Lie algebra of a Lie group in Section 5. We will compute the Lie algebras of several well known Lie groups in that section for the purpose of illustration. 2 Notation Let V be a vector space. We denote by gl(V) the space of all linear transformations on V. If V is a finite-dimensional vector space we may put an arbitrary basis on V and identify elements of gl(V) with their matrix representation. The following define various classes of matrices on Rn: ∗The University of Texas at Austin, USA. E-mail: [email protected] Last revised: July 12, 2013 Matrix Lie groups gl(n) : the space of n -
Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector
Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector Francesco de Giovanni and Carmela Musella Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy and Yaroslav P. Sysak1 Institute of Mathematics, Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine Communicated by Gernot Stroth Received November 14, 2000 DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY 1. INTRODUCTION A subgroup of a group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime order. are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 . 1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A. -
1 the Spin Homomorphism SL2(C) → SO1,3(R) a Summary from Multiple Sources Written by Y
1 The spin homomorphism SL2(C) ! SO1;3(R) A summary from multiple sources written by Y. Feng and Katherine E. Stange Abstract We will discuss the spin homomorphism SL2(C) ! SO1;3(R) in three manners. Firstly we interpret SL2(C) as acting on the Minkowski 1;3 spacetime R ; secondly by viewing the quadratic form as a twisted 1 1 P × P ; and finally using Clifford groups. 1.1 Introduction The spin homomorphism SL2(C) ! SO1;3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4 × 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1; 3). This map is alternately called the spinor map and variations. The image of this map is the identity component + of SO1;3(R), denoted SO1;3(R). The kernel is {±Ig. Therefore, we obtain an isomorphism + PSL2(C) = SL2(C)= ± I ' SO1;3(R): This is one of a family of isomorphisms of Lie groups called exceptional iso- morphisms. In Section 1.3, we give the spin homomorphism explicitly, al- though these formulae are unenlightening by themselves. In Section 1.4 we describe O1;3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct per- spectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an ex- 4 plicit action of SL2(C) on R preserving Q (Section 1.5). -
The Classical Matrix Groups 1 Groups
The Classical Matrix Groups CDs 270, Spring 2010/2011 The notes provide a brief review of matrix groups. The primary goal is to motivate the lan- guage and symbols used to represent rotations (SO(2) and SO(3)) and spatial displacements (SE(2) and SE(3)). 1 Groups A group, G, is a mathematical structure with the following characteristics and properties: i. the group consists of a set of elements {gj} which can be indexed. The indices j may form a finite, countably infinite, or continous (uncountably infinite) set. ii. An associative binary group operation, denoted by 0 ∗0 , termed the group product. The product of two group elements is also a group element: ∀ gi, gj ∈ G gi ∗ gj = gk, where gk ∈ G. iii. A unique group identify element, e, with the property that: e ∗ gj = gj for all gj ∈ G. −1 iv. For every gj ∈ G, there must exist an inverse element, gj , such that −1 gj ∗ gj = e. Simple examples of groups include the integers, Z, with addition as the group operation, and the real numbers mod zero, R − {0}, with multiplication as the group operation. 1.1 The General Linear Group, GL(N) The set of all N × N invertible matrices with the group operation of matrix multiplication forms the General Linear Group of dimension N. This group is denoted by the symbol GL(N), or GL(N, K) where K is a field, such as R, C, etc. Generally, we will only consider the cases where K = R or K = C, which are respectively denoted by GL(N, R) and GL(N, C). -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).