DOCSLIB.ORG

Sporadic and Exceptional

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sporadic and Exceptional Sporadic and Exceptional Yang-Hui He1 & John McKay2 1 Department of Mathematics, City University, London, EC1V 0HB, UK and Merton College, University of Oxford, OX14JD, UK and School of Physics, NanKai University, Tianjin, 300071, P.R. China [email protected] 2 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada [email protected] Abstract We study the web of correspondences linking the exceptional Lie algebras E8;7;6 and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of P SL(2; R) for the relevant McKay- Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle. arXiv:1505.06742v1 [math.AG] 25 May 2015 1 Contents 1 Introduction and Summary 3 2 Rudiments and Nomenclature 5 2.1 P SL(2; Z) and P SL(2; R)............................6 2.2 The Monster . 11 2.2.1 Monstrous Moonshine . 14 2.3 Exceptional Affine Lie Algebras . 15 2.4 Classical Enumerative Geometry . 18 3 Correspondences 20 3.1 Desire for Adjacency . 21 3.1.1 Initial Observation on M and Ec8 .................... 21 3.1.2 The Baby and Ec7 ............................. 22 3.1.3 Fischer and Ec6 .............................. 22 3.2 Cusp Numbers . 23 3.2.1 Cusp Character . 24 3.3 The Baby and E7 again . 28 3.4 Fischer's Group . 30 3.5 Conway's Group . 32 3.6 Genus Zero . 34 2 4 The Horrocks-Mumford Bundle: A Digression 35 4.1 Symmetry Groups . 36 4.2 The HM Quintic Calabi-Yau Threefold . 38 4.3 Embedding into Conway's Group . 40 A Character Table of GHM 42 A.1 Details of Construction . 43 1 Introduction and Summary The classification of mathematical structures oftentimes confronts the dichotomy between the regular and the exceptional wherein the former generically organizes into some infinite families whilst the latter tantalizes with what at first may seem an eclectic collage but out of whose initial disparity emerges striking order. The earliest and perhaps most well-known example is that of the classification of symmetries in R3, the regulars are the infinite families of the cyclic and dihedral groups of the regular n-gon and the exceptionals are the symmetry groups of the five Platonic solids. Similarly, the classification of Lie algebras gave us the Dynkin diagrams of the infinite families of classical groups which are the familiar isometries of vector spaces, in addition, there are the five exceptional diagrams. Indeed, the relation between the R3 symmetries and the simply laced cases of the (affine) Lie algebras has come to be known as the McKay Correspondence, which now occupies a cornerstone of modern algebraic geometry and representation theory. Another highlight example is of course the classification of finite simple groups which, after decades of arduous work, is by now complete. The regulars here are the infinite families 3 of cyclic groups of prime order, as well as the classical Lie groups over finite fields, while the exceptionals are known as the 26 sporadic groups, the largest of which is the Monster, of tremendous size. Here, the second author's old observation that 196; 884 = 196; 883 + 1, relating the Monster and the elliptic j-function, prompted the field of Moonshine [1,28]. These above illustrative cases indeed exemplify how astute recognition of exceptional structures can help to unravel new mathematics of profound depth. Of particular curiosity is the less well-known fact - in parallel to the above identity - that the constant term of the j- invariant, viz., 744, satisfies 744 = 3×248. The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra E8. In fact, that j should encode the representations of E8 was settled [41] long before the final proof of the Moonshine conjectures [29]. This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics. Over the years, there have been various generalizations of Monstrous Moonshine (cf. [7]) in mathematics and, more recently, in physics [16{20]. Of the vast literature, we will focus on relating the exceptional algebras and the sporadics, review the pertinent background and present a new set of correspondences, which though numeralogically intriguing, remain fundamentally mysterious and await further exploration. To guide the reader, we summarize the web of inter-connections in Figure 1 as well as the ensuing list which point to the relevant sections: • McKay Correspondence between Eb6;7;8 Dynkin diagrams and symmetries of Platonic solids in (2.25) and (2.26); • Dimensions of fundamental representations of E6;7;8 in (2.29), number of lines (respec- tively bitangents and tritangents) as well as (−1)-curves in classical geometry of del Pezzo surfaces and curves thereon in Table 1; • Classes of involutions in M and the nodes of the Ec8 Dynkin diagram in (3.5), similarly, 0 involutory classes in 2:B and Ec7 in (3.6), and 3-transposition classes of 3:F i24 and Ec6 in (3.7); 0 • Cusp number sums of the invariant groups of McKay-Thompson series for M; 2:B; 3:F i24 and the enumerative geometries for the del Pezzo surfaces dP3;2;1 in Observations 1, 2, and 3; 4 Figure 1: The web of correspondences amongst the exceptional Lie algebras and sporadic groups, via some classical enumerative geometry and modular subgroups. • Conway's group Co1 and the Horrocks-Mumford bundle in Observation 5. The outline of the paper is as follows. We begin with setting the notation for the various subjects upon which we will touch, as well as reviewing the rudiments in as a self-contained manner as possible in x2, before delving into our correspondences in x3. We end with a digression on the Horrocks-Mumford bundle in x4. 2 Rudiments and Nomenclature First let us set the notation, and refresh the reader's mind on the characteristics of the various dramatis personae. 5 2.1 P SL(2; Z) and P SL(2; R) We will denote the modular group as ! a b Γ := P SL(2; Z) = f j a; b; c; d 2 Z; ad − bc = 1g={±Ig : (2.1) c d The two important subgroups of Γ for our purposes are the congruence subgroups: Γ0(N) := fγ 2 Γ j c mod N = 0g ; ! a b Γ(N) := fγ 2 Γ j mod N = Ig ⊂ Γ0(N) : (2.2) c d Cusps: The action of Γ on the upper-half plane H = fz : Im(z) > 0g by linear-fractional az+b transformation z 7! cz+d is fundamental. In addition to H, of particular interest is the set of cusps Q [ f1g on the boundary real axis. These are rational points which are taken to themselves under the linear-fractional maps in Γ and are the only points on the real axis which should be adjoined to H when considering the action of Γ. Thus, we will generally speak of the extended upper half plane ∗ H := H[ Q [ f1g : (2.3) For any subgroup Θ of Γ (including itself), we can define the set of cusps as C(Θ) := fΘ-orbits of Q [ f1gg: (2.4) An important fact about congruence subgroups Θ is that C(Θ) is finite1 and we will hence- forth call the number of elements in C(Θ) the cusp number. Indeed, with respect to the full modular group, for any two points in the extended upper half-plane 2 x; y 2 Q[f1g, there exists a γ 2 Γ such that γ(x) = y. Thus, the cusp number of the full modular group is 1, i.e., jC(Γ)j= 1. 1 As above, we can show this using representatives modulo the level of the congruence and see the image of x = px . qx 2 This can be easily shown by writing, as explicit fractions, x = px and y = py and solving for integer qx qy a px +b py qx a; b; c; d such that q = c px +d and ad − bc = 1. y qx 6 Q 1 For the Hecke groups, the index of Γ0(N) in Γ is N 1 + p and pjN X jC(Γ0(N)j= φ(gcd(d; N=d)) ; (2.5) djN;d>0 Q 1 where φ(m) = m 1 − p is the standard Euler totient counting the number of positive pjm integers between 1 and m, inclusive, which are co-prime to m. Modular Curves: By adjoining appropriate cusps which serve as compactification points, we can form the quotient ΘnH∗ of the extended upper half plane by any subgroups Θ of Γ. The first classical result 3 , dating to Klein and Dedekind, is that for the full modular group Γ, this quotient is the Riemann sphere: ∗ 1 ΓnH ' P : (2.6) In general, we can quotient H∗ by a congruence subgroup and obtain a Riemann surface, dubbed the modular curve. Commonly, we denote the modular curves associated to Γ, Γ0(N) 1 and Γ(N) respectively as X(Γ) ' P , X0(N) and X(N). What subgroups Θ also have X(Θ) ' P1? This so-called genus zero property is important in several contexts and subgroups possessing it are quite rare. For example, that there are only 33 (finite-index) torsion-free genus zero subgroups was the classification of [11] and the physical interpretation, the subject of [20,24].
Load more

Recommended publications
  • Near Polygons and Fischer Spaces
    Near polygons and Fischer spaces Citation for published version (APA): Brouwer, A. E., Cohen, A. M., Hall, J. I., & Wilbrink, H. A. (1994). Near polygons and Fischer spaces. Geometriae Dedicata, 49(3), 349-368. https://doi.org/10.1007/BF01264034 DOI: 10.1007/BF01264034 Document status and date: Published: 01/01/1994 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • 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.
    [Show full text]
  • Quadratic Forms and Their Applications
    Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999.
    [Show full text]
  • 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.
    [Show full text]
  • THE FISCHER-CLIFFORD MATRICES of a MAXIMAL SUBGROUP of Fi24 1. Introduction Let ¯G = N·G Be an Extension of N by G, Where N Is
    REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 7, Pages 300{321 (July 29, 2003) S 1088-4165(03)00175-4 THE FISCHER-CLIFFORD MATRICES 0 OF A MAXIMAL SUBGROUP OF Fi24 FARYAD ALI AND JAMSHID MOORI Abstract. 0 The Fischer group Fi24 is the largest sporadic simple Fischer group of order 1255205709190661721292800 = 221:316:52:73:11:13:17:23:29 : 0 The group Fi24 is the derived subgroup of the Fischer 3-transposition group Fi24 discovered by Bernd Fischer. There are five classes of elements of order 0 3inFi24 as represented in ATLAS by 3A,3B,3C,3D and 3E. A subgroup 0 2f g of Fi24 of order 3 is called of type 3X,whereX A; B; C; D; E ,ifitisgen- erated by an element in the class 3X. There are six classes of maximal 3-local 0 subgroups of Fi24 as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local ¯ h i ∼ 7· ∼ 7 subgroups G := NFi0 ( N ) = 3 O7(3), where N = 3 is the natural orthog- 24∼ onal module for G=N¯ = O7(3) with 364 subgroups of type 3B corresponding to the totally isotropic points. The group G¯ is a nonsplit extension of N by ∼ G = O7(3). 1. Introduction Let G¯ = N·G be an extension of N by G,whereN is a normal subgroup of G¯ ∼ and G=N¯ = G.Let¯g 2 G¯ be a lifting of g 2 G under the natural homomorphism G¯ −→ G and let [g] be a conjugacy class of elements of G with representative g.
    [Show full text]
  • 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.
    [Show full text]
  • THERE IS NO Sz(8) in the MONSTER 1. Introduction The
    THERE IS NO Sz(8) IN THE MONSTER ROBERT A. WILSON Abstract. As a contribution to an eventual solution of the problem of the determination of the maximal subgroups of the Monster we show that there is no subgroup isomorphic to Sz(8). This also completes the determination of exactly which simple groups are subgroups of which of the 26 sporadic simple groups. The proof is largely, though not entirely, computer-free. 1. Introduction The Fischer{Griess Monster group M is the largest of the 26 sporadic simple groups, and was first constructed by Griess [6] in 1982. A simplified construction along the same general lines was given by Conway [2]. One of the major problems in group theory today is that of classifying the max- imal subgroups of the finite simple groups and their automorphism groups (see, for example, [1]). Much work has been done over many years attempting to determine the maximal subgroups of M, but it is still the only sporadic group whose maximal subgroups are not completely classified (see [23] and references therein). The maximal p-local subgroups of the Monster were classified in [22, 15, 16], and much theoretical work on non-local subgroups was accomplished in [17, 18]. Following successful computer constructions of the Monster [14, 8] other techniques became available, and further progress was made [9, 11, 7, 19, 26, 27], including discovery of five previously unknown maximal subgroups, isomorphic to PSL2(71), PSL2(59), PSL2(41), PGL2(29), PGL2(19). The cases left open by this previous work are possible maximal subgroups with socle isomorphic to one of the following simple groups: PSL2(8); PSL2(13); PSL2(16); PSU3(4); PSU3(8); Sz(8): Of these, PSL2(8) and PSL2(16) have been classified in unpublished work of P.
    [Show full text]
  • The Symmetric Genus of the Fischer Group Fi23
    The symmetric genus of the Fischer group F i23 Robert A. Wilson School of Mathematics and Statistics, The University of Birmingham, England published in Topology 36 (1996), 379{380 Abstract We show that the sporadic simple group F i23 is generated by an element of order 2 and an element of order 3, whose product has order 8. Since F i23 is not a Hurwitz group, we can deduce the symmetric genus of the group. In [4] the symmetric genus of a finite group G is defined to be the smallest integer g such that G acts faithfully on a closed orientable surface of genus g. The Riemann{Hurwitz formula implies that this minimum is attained by minimising n 1 X 1 − i=1 ai n Q ai where G is generated by elements xi satisfying xj = 1 and xi = 1 for all j=1 i. In most, but not all, cases, this minimum is attained for n = 3, and we have 1 1 1 1 g = 1 + jGj(1 − − − ); 2 r s t where 1 1 1 + + r s t is maximal subject to the existence of elements x, y and z generating G, with xr = ys = zt = xyz = 1: 1 The symmetric genus of each of the sporadic simple groups is discussed in [2], where a table of results shows that the symmetric genus is now known in 23 of the 26 cases. The remaining three are F i23, B and M. The case of the Baby Monster was dealt with in [5]. In this paper we deal with the case F i23.
    [Show full text]
  • The Leech Lattice Λ and the Conway Group ·O Revisited
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 3, March 2010, Pages 1351–1369 S 0002-9947(09)04726-6 Article electronically published on October 20, 2009 THE LEECH LATTICE Λ AND THE CONWAY GROUP ·O REVISITED JOHN N. BRAY AND ROBERT T. CURTIS Dedicated to John Horton Conway as he approaches his seventieth birthday. Abstract. We give a new, concise definition of the Conway group ·Oasfol- lows. The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of 24 tetrads. We use this 4 24 action to define a progenitor P of shape 2 4 :M24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conway’s elements ξT and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under ·O of the integral vectors in R24. 1. Introduction The Leech lattice Λ was discovered by Leech [14] in 1965 in connection with the packing of non-overlapping identical spheres into the 24–dimensional space R24 so that their centres lie at the lattice points; see Conway and Sloane [9]. Its con- struction relies heavily on the rich combinatorial structure underlying the Mathieu group M24. The full group of symmeries of Λ is, of course, infinite, as it contains all translations by a lattice vector.
    [Show full text]
  • Finite Groups Acting on Homology Manifolds
    pacific journal of mathematics Vol. 181, No. 3, 1997 FINITE GROUPS ACTING ON HOMOLOGY MANIFOLDS Michael Aschbacher To the memory of Olga Taussky-Todd 1. Introduction. In this paper we study homology manifolds T admitting the action of a fi- nite group preserving the structure of a regular CW-complex on T . The CW-complex is parameterized by a poset and the topological properties of the manifold are translated into a combinatorial setting via the poset. We concentrate on n-manifolds which admit a fairly rigid group of automor- phisms transitive on the n-cells of the complex. This allows us to make yet another translation from a combinatorial into a group theoretic setting. We close by using our machinery to construct representations on manifolds of the Monster, the largest sporadic group. Some of these manifolds are of dimension 24, and hence candidates for examples to Hirzebruch’s Prize Question in [HBJ], but unfortunately closer inspection shows the Aˆ-genus of these manifolds is 0 rather than 1, so none is a Hirzebruch manifold. In order to state our results precisely we need to recall some definitions and introduce a few concepts. The geometric realization functor T associates to each poset X a topological space T (X) and to each x ∈ X a closed subspace T (x)ofT(X). The cells T (x), x ∈ X, supply a cell structure on T (X) parametrized by X. In addition there is a simplicial complex O(X) associated to X called the order complex of X and a canonical triangulation ϕ : O(X) → T (X)ofT(X)byO(X).
    [Show full text]
  • Arxiv:1211.5531V2 [Math.RT] 15 Mar 2013 Much Ado About Mathieu
    Much ado about Mathieu Terry Gannon Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 e-mail: [email protected] March 18, 2013 Abstract Eguchi, Ooguri and Tachikawa have observed that the elliptic genus of type II string theory on K3 surfaces appears to possess a Moonshine for the largest Mathieu group. Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M24. In this paper we prove that the resulting sequence of class functions are true characters of M24, proving the Eguchi-Ooguri-Tachikawa conjecture. The integrality of multiplicities is proved using a small generalisation of Sturm’s Theorem, while positivity in- volves a modification of a method of Hooley. We also prove the evenness prop- erty of the multiplicities, as conjectured by several authors. We also identify the role group cohomology plays in both K3-Mathieu Moonshine and Mon- strous Moonshine; in particular this gives a cohomological interpretation for the non-Fricke elements in Norton’s Generalised Monstrous Moonshine con- jecture. We investigate the proposal of Gaberdiel-Hohenegger-Volpato that K3-Mathieu Moonshine lifts to the Conway group Co1. Contents arXiv:1211.5531v2 [math.RT] 15 Mar 2013 1 Introduction 2 2 K3-Mathieu Moonshine: Review 5 3 Weak K3-Mathieu Moonshine I: Integrality 11 4 Weak Mathieu Moonshine II: Positivity 17 5 Is the Conway group the stringy symmetry? 30 6 Speculations 33 References 37 1 1 Introduction The elliptic genus (a.k.a. partition function) of a nonlinear sigma model with K3 target space is a very special function.
    [Show full text]
  • Words for Maximal Subgroups of Fi '
    Open Chem., 2019; 17: 1491–1500 Research Article Faisal Yasin, Adeel Farooq, Chahn Yong Jung* Words for maximal Subgroups of Fi24‘ https://doi.org/10.1515/chem-2019-0156 received October 16, 2018; accepted December 1, 2019. energy levels, and even bond order to name a few can be found, all without rigorous calculations [2]. The fact that Abstract: Group Theory is the mathematical application so many important physical aspects can be derived from of symmetry to an object to obtain knowledge of its symmetry is a very profound statement and this is what physical properties. The symmetry of a molecule provides makes group theory so powerful [3,4]. us with the various information, such as - orbitals energy The allocated point groups would then be able to levels, orbitals symmetries, type of transitions than can be utilized to decide physical properties, (for example, occur between energy levels, even bond order, all that concoction extremity and chirality), spectroscopic without rigorous calculations. The fact that so many properties (especially valuable for Raman spectroscopy, important physical aspects can be derived from symmetry infrared spectroscopy, round dichroism spectroscopy, is a very profound statement and this is what makes mangnatic dichroism spectroscopy, UV/Vis spectroscopy, group theory so powerful. In group theory, a finite group and fluorescence spectroscopy), and to build sub-atomic is a mathematical group with a finite number of elements. orbitals. Sub-atomic symmetry is in charge of numerous A group is a set of elements together with an operation physical and spectroscopic properties of compounds and which associates, to each ordered pair of elements, an gives important data about how chemical reaction happen.
    [Show full text]