The Leech Lattice Λ and the Conway Group ·O Revisited
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Generating the Mathieu Groups and Associated Steiner Systems
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 112 (1993) 41-47 41 North-Holland Generating the Mathieu groups and associated Steiner systems Marston Conder Department of Mathematics and Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand Received 13 July 1989 Revised 3 May 1991 Abstract Conder, M., Generating the Mathieu groups and associated Steiner systems, Discrete Mathematics 112 (1993) 41-47. With the aid of two coset diagrams which are easy to remember, it is shown how pairs of generators may be obtained for each of the Mathieu groups M,,, MIz, Mz2, M,, and Mz4, and also how it is then possible to use these generators to construct blocks of the associated Steiner systems S(4,5,1 l), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) respectively. 1. Introduction Suppose you land yourself in 24-dimensional space, and are faced with the problem of finding the best way to pack spheres in a box. As is well known, what you really need for this is the Leech Lattice, but alas: you forgot to bring along your Miracle Octad Generator. You need to construct the Leech Lattice from scratch. This may not be so easy! But all is not lost: if you can somehow remember how to define the Mathieu group Mz4, you might be able to produce the blocks of a Steiner system S(5,8,24), and the rest can follow. In this paper it is shown how two coset diagrams (which are easy to remember) can be used to obtain pairs of generators for all five of the Mathieu groups M,,, M12, M22> Mz3 and Mz4, and also how blocks may be constructed from these for each of the Steiner systems S(4,5,1 I), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) respectively. -
Some Design Theoretic Results on the Conway Group ·0
Some design theoretic results on the Conway group ·0 Ben Fairbairn School of Mathematics University of Birmingham, Birmingham, B15 2TT, United Kingdom [email protected] Submitted: Oct 3, 2008; Accepted: Jan 14, 2010; Published: Jan 22, 2010 Mathematics Subject Classification: 05B99, 05E10, 05E15, 05E18 Abstract Let Ω be a set of 24 points with the structure of the (5,8,24) Steiner system, S, defined on it. The automorphism group of S acts on the famous Leech lattice, as does the binary Golay code defined by S. Let A, B ⊂ Ω be subsets of size four (“tetrads”). The structure of S forces each tetrad to define a certain partition of Ω into six tetrads called a sextet. For each tetrad Conway defined a certain automorphism of the Leech lattice that extends the group generated by the above to the full automorphism group of the lattice. For the tetrad A he denoted this automorphism ζA. It is well known that for ζA and ζB to commute it is sufficient to have A and B belong to the same sextet. We extend this to a much less obvious necessary and sufficient condition, namely ζA and ζB will commute if and only if A∪B is contained in a block of S. We go on to extend this result to similar conditions for other elements of the group and show how neatly these results restrict to certain important subgroups. 1 Introduction The Leech lattice, Λ, was discovered by Leech in 1965 in connection with the packing of spheres into 24-dimensional space R24, so that their centres form a lattice. -
Sphere Packing, Lattice Packing, and Related Problems
Sphere packing, lattice packing, and related problems Abhinav Kumar Stony Brook April 25, 2018 Sphere packings Definition n A sphere packing in R is a collection of spheres/balls of equal size which do not overlap (except for touching). The density of a sphere packing is the volume fraction of space occupied by the balls. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem II In dimension 3, the best possible way is to stack layers of the solution in 2 dimensions. This is Kepler's conjecture, now a theorem of Hales and collaborators. mmm m mmm m There are infinitely (in fact, uncountably) many ways of doing this! These are the Barlow packings. Face centered cubic packing Image: Greg A L (Wikipedia), CC BY-SA 3.0 license But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered. -
ORBITS in the LEECH LATTICE 1. Introduction the Leech Lattice Λ Is a Lattice in 24-Dimensional Euclidean Space with Many Remark
ORBITS IN THE LEECH LATTICE DANIEL ALLCOCK Abstract. We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, 18 the Conway group Co 0 of order 8 10 . Our methods rely on and develop the work of R. T.∼ Curtis,× and we describe our in- tended applications to the symmetry groups of Lorentzian lattices and the enumeration of lattices of dimension 24 with good prop- erties such as having small determinant. Our≈ algorithm reduces 12 the test of equivalence to 4 tests under the subgroup 2 :M24, ≤ and a test under this subgroup to 12 tests under M24. We also give algorithms for testing equivalence≤ under these two subgroups. Finally, we analyze the performance of the algorithm. 1. Introduction The Leech lattice Λ is a lattice in 24-dimensional Euclidean space with many remarkable properties, for us the most important of which is that its isometry group (modulo I ) is one of the sporadic finite {± } simple groups. The isometry group is called the Conway group Co 0, and our purpose is to present an algorithm for a computer to determine whether two given vectors of Λ are equivalent under Co 0. The group is fairly large, or order > 8 1018, so the obvious try-every-isometry algorithm is useless. Instead,× we use the geometry of Λ to build a faster algorithm; along the way we build algorithms for the problem 24 12 of equivalence of vectors in R under the subgroups 2 :M24 and M24, where M24 is the Mathieu group permuting the 24 coordinates. -
Optimality and Uniqueness of the Leech Lattice Among Lattices
ANNALS OF MATHEMATICS Optimality and uniqueness of the Leech lattice among lattices By Henry Cohn and Abhinav Kumar SECOND SERIES, VOL. 170, NO. 3 November, 2009 anmaah Annals of Mathematics, 170 (2009), 1003–1050 Optimality and uniqueness of the Leech lattice among lattices By HENRY COHN and ABHINAV KUMAR Dedicated to Oded Schramm (10 December 1961 – 1 September 2008) Abstract We prove that the Leech lattice is the unique densest lattice in R24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R24 can 30 exceed the Leech lattice’s density by a factor of more than 1 1:65 10 , and we 8 C give a new proof that E8 is the unique densest lattice in R . 1. Introduction It is a long-standing open problem in geometry and number theory to find the densest lattice in Rn. Recall that a lattice ƒ Rn is a discrete subgroup of rank n; a minimal vector in ƒ is a nonzero vector of minimal length. Let ƒ vol.Rn=ƒ/ j j D denote the covolume of ƒ, i.e., the volume of a fundamental parallelotope or the absolute value of the determinant of a basis of ƒ. If r is the minimal vector length of ƒ, then spheres of radius r=2 centered at the points of ƒ do not overlap except tangentially. This construction yields a sphere packing of density n=2 r Án 1 ; .n=2/Š 2 ƒ j j since the volume of a unit ball in Rn is n=2=.n=2/Š, where for odd n we define .n=2/Š .n=2 1/. -
Octonions and the Leech Lattice
Octonions and the Leech lattice Robert A. Wilson School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS 18th December 2008 Abstract We give a new, elementary, description of the Leech lattice in terms of octonions, thereby providing the first real explanation of the fact that the number of minimal vectors, 196560, can be expressed in the form 3 × 240 × (1 + 16 + 16 × 16). We also give an easy proof that it is an even self-dual lattice. 1 Introduction The Leech lattice occupies a special place in mathematics. It is the unique 24- dimensional even self-dual lattice with no vectors of norm 2, and defines the unique densest lattice packing of spheres in 24 dimensions. Its automorphism group is very large, and is the double cover of Conway’s group Co1 [2], one of the most important of the 26 sporadic simple groups. This group plays a crucial role in the construction of the Monster [13, 4], which is the largest of the sporadic simple groups, and has connections with modular forms (so- called ‘Monstrous Moonshine’) and many other areas, including theoretical physics. The book by Conway and Sloane [5] is a good introduction to this lattice and its many applications. It is not surprising therefore that there is a huge literature on the Leech lattice, not just within mathematics but in the physics literature too. Many attempts have been made in particular to find simplified constructions (see for example the 23 constructions described in [3] and the four constructions 1 described in [15]). -
The Moonshine Module for Conway's Group
Forum of Mathematics, Sigma (2015), Vol. 3, e10, 52 pages 1 doi:10.1017/fms.2015.7 THE MOONSHINE MODULE FOR CONWAY’S GROUP JOHN F. R. DUNCAN and SANDER MACK-CRANE Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA; email: [email protected], [email protected] Received 26 September 2014; accepted 30 April 2015 Abstract We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise. 2010 Mathematics Subject Classification: 11F11, 11F22, 17B69, 20C34 1. Introduction Taking the upper half-plane H τ C (τ/ > 0 , together with the Poincare´ 2 2 2 2 VD f 2 j = g metric ds y− .dx dy /, we obtain the Poincare´ half-plane model of the D C hyperbolic plane. -
A Uniform Construction of the Holomorphic Voas of Central
A Uniform Construction of the 71 Holomorphic VOAs of c = 24 from the Leech Lattice Sven Möller (joint work with Nils Scheithauer) Vertex Operator Algebras, Number Theory, and Related Topics: A Conference in Honor of Geoffrey Mason, California State University, Sacramento 12th June 2018 Schellekens’ List Orbifold Construction and Dimension Formulae A Generalised Deep-Hole Construction Contents 1 Schellekens’ List 2 Orbifold Construction and Dimension Formulae 3 A Generalised Deep-Hole Construction Sven Möller A Uniform Construction of the 71 holomorphic VOAs of c = 24 Schellekens’ List Orbifold Construction and Dimension Formulae A Generalised Deep-Hole Construction Holomorphic VOAs of Small Central Charge Proposition (Consequence of [Zhu96]) Let V be a strongly rational, holomorphic VOA. Then the central charge c of V is in 8Z≥0. + c = 8: VE8 , c = 16: V 2 , V (only lattice theories) [DM04] E8 D16 Theorem ([Sch93, DM04, EMS15]) Let V be a strongly rational, holomorphic VOA of central charge c = 24. Then the Lie algebra V1 is isomorphic to one of the 71 Lie algebras on Schellekens’ list (V \, 24 Niemeier lattice theories, etc. with chV (τ) = j(τ) − 744 + dim(V1)). c = 32: already more than 1 160 000 000 lattice theories Sven Möller A Uniform Construction of the 71 holomorphic VOAs of c = 24 Schellekens’ List Orbifold Construction and Dimension Formulae A Generalised Deep-Hole Construction Classification Orbifold constructions give all 71 cases on Schellekens’ list. [FLM88, DGM90, Don93, DGM96, Lam11, LS12, LS15, Miy13, SS16, EMS15, Mö16, LS16b, LS16a, LL16] Theorem (Classification I) There is a strongly rational, holomorphic VOA V of central charge c = 24 with Lie algebra V1 if and only if V1 is isomorphic to one of the 71 Lie algebras on Schellekens’ list. -
Error-Correction and the Binary Golay Code
London Mathematical Society Impact150 Stories 1 (2016) 51{58 C 2016 Author(s) doi:10.1112/i150lms/t.0003 e Error-correction and the binary Golay code R.T.Curtis Abstract Linear algebra and, in particular, the theory of vector spaces over finite fields has provided a rich source of error-correcting codes which enable the receiver of a corrupted message to deduce what was originally sent. Although more efficient codes have been devised in recent years, the 12- dimensional binary Golay code remains one of the most mathematically fascinating such codes: not only has it been used to send messages back to earth from the Voyager space probes, but it is also involved in many of the most extraordinary and beautiful algebraic structures. This article describes how the code can be obtained in two elementary ways, and explains how it can be used to correct errors which arise during transmission 1. Introduction Whenever a message is sent - along a telephone wire, over the internet, or simply by shouting across a crowded room - there is always the possibly that the message will be damaged during transmission. This may be caused by electronic noise or static, by electromagnetic radiation, or simply by the hubbub of people in the room chattering to one another. An error-correcting code is a means by which the original message can be recovered by the recipient even though it has been corrupted en route. Most commonly a message is sent as a sequence of 0s and 1s. Usually when a 0 is sent a 0 is received, and when a 1 is sent a 1 is received; however occasionally, and hopefully rarely, a 0 is sent but a 1 is received, or vice versa. -
Quaternions Over Q(&)
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOCRNAL OF ALGEBRA 63, 56-75 (I 980) Quaternions over Q(&), Leech’s Lattice and the Sporadic Group of Hall-lank0 J. TITS CollSge de France, I1 Place Marc&n-Berthelot, 75231 Paris Cedex 0.5, France Communicated by I. N. Herstein Received April 24, 1979 TO MY FRIEND NATHAN JACOBSON ON HIS 70TH BIRTHDAY INTRODUCTION It was observed long ago by J. Thompson that the automorphism group of Leech’s lattice A-Conway’s group *O-contains a subgroup X, isomorphic to the double cover &, of the alternating group ‘SI, . Furthermore, if we denote by X,3X,3 **. r) Xs the decreasing sequence of subgroups of Xs charac- terized up to conjugacy by Xi g ‘& , the centralizers Ci = C.,(XJ form a remarkable sequence: C, = .O is the double cover of Conway’s group ‘1, Cs the sextuple cover c, of Suzuki’s sporadic group, C, the double cover of Ga(F,), C’s the double cover J, of the sporadic group of Hall-Janko, . (cf. [4, p. 2421). This suggests a variety of interesting ways of looking at A, namely, as a module over its endomorphism ring Ri generated by Xi , for i = 2, 3, 4,...: indeed, the automorphism group of that module (endowed with a suitable form) is precisely the group Ci . For i = 2, R, = Z and one simply has Leech’s and Conway’s original approaches to the lattice [2, 10, 111. For i = 3, Rd = Z[$‘i] and A appears as the R,-module of rank 12 investigated by Lindsey [13] (cf. -
Grundlehren Der Mathematischen Wissenschaften 290 a Series of Comprehensive Studies in Mathematics
Grundlehren der mathematischen Wissenschaften 290 A Series of Comprehensive Studies in Mathematics Editors S.S. Chern B. Eckmann P. de la Harpe H. Hironaka F. Hirzebruch N. Hitchin L. Hormander M.-A. Knus A. Kupiainen J. Lannes G. Lebeau M. Ratner D. Serre Ya.G. Sinai N. J. A. Sloane J. Tits M. Waldschmidt S. Watanabe Managing Editors M. Berger J. Coates S.R.S. Varadhan Springer Science+Business Media, LLC J.H. Conway N .J.A. Sloane Sphere Packings, Lattices and Groups Third Edition With Additional Contributions by E. Bannai, R.E. Borcherds, J. Leech, S.P. Norton, A.M. Odlyzko, R.A. Parker, L. Queen and B. B. Venkov With 112 Illustrations 'Springer J.H. Conway N.J.A. Sloane Mathematics Department Information Sciences Research Princeton University AT&T Labs - Research Princeton, NJ 08540 USA 180 Park Avenue conway@ math. princeton.edu Florham Park, NJ 07932 USA [email protected] Mathematics Subject Classification (1991): 05B40, 11H06, 20E32, 11T71, 11E12 Library of Congress Cataloging-in-Publication Data Conway, John Horton Sphere packings, lattices and groups.- 3rd ed./ J.H. Conway, N.J.A. Sloane. p. em. - (Grundlehren der mathematischen Wissenschaften ; 290) Includes bibliographical references and index. ISBN 978-1-4419-3134-4 ISBN 978-1-4757-6568-7 (eBook) DOI 10.1007/978-1-4757-6568-7 1. Combinatorial packing and covering. 2. Sphere. 3. Lattice theory. 4. Finite groups. I. Sloane, N.J.A. (Neil James Alexander). 1939- . II. Title. III. Series. QA166.7.C66 1998 511'.6-dc21 98-26950 Printed on acid-free paper. springeronline.com © 1999, 1998, 1993 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. -
Octonions and the Leech Lattice
Journal of Algebra 322 (2009) 2186–2190 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Octonions and the Leech lattice Robert A. Wilson School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom article info abstract Article history: We give a new, elementary, description of the Leech lattice in Received 18 December 2008 terms of octonions, thereby providing the first real explanation Availableonline27March2009 of the fact that the number of minimal vectors, 196560, can be Communicated by Gernot Stroth expressed in the form 3 × 240 × (1 + 16 + 16 × 16).Wealsogive an easy proof that it is an even self-dual lattice. Keywords: © Octonions 2009 Elsevier Inc. All rights reserved. Leech lattice Conway group 1. Introduction The Leech lattice occupies a special place in mathematics. It is the unique 24-dimensional even self-dual lattice with no vectors of norm 2, and defines the unique densest lattice packing of spheres in 24 dimensions. Its automorphism group is very large, and is the double cover of Conway’s group Co1 [2], one of the most important of the 26 sporadic simple groups. This group plays a crucial role in the construction of the Monster [14,4], which is the largest of the sporadic simple groups, and has connections with modular forms (so-called ‘Monstrous Moonshine’) and many other areas, including theoretical physics. The book by Conway and Sloane [5] is a good introduction to this lattice and its many applications. It is not surprising therefore that there is a huge literature on the Leech lattice, not just within mathematics but in the physics literature too.