Monster Group Research Articles

Higher index theory for certain expanders and Gromov monster groups, II

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum–Connes assembly map is an isomorphism for the associated metric space X. As discussed in the first paper in this series, this has applications to the Baum–Connes conjecture for ‘Gromov monster’ groups. We also introduce a new property, ‘geometric property (T)’. For the metric space associated to a sequence of graphs, this property is an obstruction to the maximal coarse assembly map being an isomorphism. This enables us to distinguish between expanders with girth tending to infinity, and, for example, those constructed from property (T) groups.

Higher index theory for certain expanders and Gromov monster groups, I

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum–Connes assembly map is injective, but not surjective, for the associated metric space X. Expanders with this girth property are a necessary ingredient in the construction of the so-called ‘Gromov monster’ groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum–Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum–Connes assembly map with these coefficients is an isomorphism.

Majorana representations of A 5

The Monster group M, which is the largest among the 26 sporadic simple groups is the automorphism group of the 196,884-dimensional Conway–Griess–Norton algebra (simply called the Monster algebra). There is a remarkable correspondence between the so-called 2A-involutions in M and certain idempotents in the Monster algebra (we refer to these idempotents as Majorana axes). The isomorphism types of the subalgebras in the Monster algebra generated by pairs of Majorana axes were calculated by S. Norton a while ago (there are precisely nine isomorphism types). More recently these nine algebras were characterized by S. Sakuma in the context of Vertex Operator Algebras, relying on earlier work by M. Miyamoto. The properties of Monster algebras used in the proof of Sakuma’s theorem are rather elementary and they have been axiomatized under the name of Majorana representations. In this terminology Sakuma’s theorem amounts to classification of the Majorana representations of the dihedral groups together with a remark that all the representations are based on embeddings into the Monster. In the present paper it is shown that the alternating group A 5 of degree 5 possesses precisely two Majorana representations, both based on embeddings into the Monster. The dimensions of the representations are 20 and 26; the scalar squares of their identities are 10 and 72/7, respectively (in the Vertex Operator Algebra context these numbers are doubled central charges).

On Majorana Representations of A 6 and A 7

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009) by axiomatising some properties of the 2A-axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2A-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to G the action of the Monster on its algebra. This representation of G is said to be based on an embedding of G in the Monster. So far the Majorana representations have been classified for the groups G isomorphic to the symmetric group S 4 of degree 4 (Ivanov et al. in J Alg 324:2432–2463, 2010), the alternating group A 5 of degree 5 (Ivanov AA, Seress A in Majorana Representations of A 5, 2010), and the general linear group GL 3(2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L 3(2), 2010). All these representations are based on embeddings in the Monster of either the group G itself or of its direct product with a cyclic group of order 2. The dimensions and shapes of these representations are given in the following table:

Orbit invariants and an application to the Baby monster

We prove that the minimal base size for the permutation action of the sporadic simple Baby monster group B on the cosets of its 7th and 8th maximal subgroup (in decreasing order of size) is 3 and 2 respectively. Motivated by the large sizes of these permutation actions, we develop new computational methods to prove that an orbit is regular and to show that two orbits are disjoint.

Introduction to Sporadic Groups

This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the $1+1+16=18$ families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated "pariah" groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group $\mathbb M$, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the $5+7+8+6=26$ sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.

Majorana representations of the symmetric group of degree 4

The Monster group M acts on a real vector space VM of dimension 196,884 which is the sum of a trivial 1-dimensional module and a minimal faithful M-module. There is an M-invariant scalar product (,) on VM, an M-invariant bilinear commutative non-associative algebra product ⋅ on VM (commonly known as the Conway–Griess–Norton algebra), and a subset A of VM∖{0} indexed by the 2A-involutions in M. Certain properties of the quintetM=(M,VM,A,(,),⋅) have been axiomatized in Chapter 8 of Ivanov (2009) [Iv09] under the name of Majorana representation of M. The axiomatization enables one to study Majorana representations of an arbitrary group G (generated by its involutions). A representation might or might not exist, but it always exists whenever G is a subgroup in M generated by the 2A-involutions contained in G. We say that thus obtained representation is based on an embedding of G in the Monster. The essential motivation for introducing the Majorana terminology was the most remarkable result by S. Sakuma (2007) [Sak07] which gave a classification of the Majorana representations of the dihedral groups. There are nine such representations and every single one is based on an embedding in the Monster of the relevant dihedral group. It is a fundamental property of the Monster that its 2A-involutions form a class of 6-transpositions and that there are precisely nine M-orbits on the pairs of 2A-involutions (and also on the set of 2A-generated dihedral subgroups in M). In the present paper we are making a further step in building up the Majorana theory by classifying the Majorana representations of the symmetric group S4 of degree 4. We prove that S4 possesses precisely four Majorana representations. The Monster is known to contain four classes of 2A-generated S4-subgroups, so each of the four representations is based on an embedding of S4 in the Monster. The classification of 2A-generated S4-subgroups in the Monster relies on calculations with the character table of the Monster. Our elementary treatment shows that there are (at most) four isomorphism types of subalgebras in the Conway–Griess-Norton algebra of the Monster generated by six Majorana axial vectors canonically indexed by the transpositions of S4. Two of these subalgebras are 13-dimensional, the other two have dimensions 9 and 6. These dimensions, not to mention the isomorphism type of the subalgebras, were not known before.

Base sizes for sporadic simple groups

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) ⩽ 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.

Geometry and Arithmetic around Hypergeometric Functions

The workshop Geometry and Arithmetic around Hypergeometric Functions , organised by Gert Heckman (Nijmegen), Jürgen Wolfart (Frankfurt a.M.) and Masaaki Yoshida (Fukuoka) was held September 28th – October 4th, 2008. It was attended by more than 40 participants coming from more than 10 different countries and representing a remarkable variety of mathematical research directions centered around hypergeometric functions. The public of the conference was mixed not only with respect to its mathematical interests but also in age and experience: it ranged from doctoral students to excellent and well-known senior researchers. The program concentrated on important new developements of the last years, let sufficient time for discussions in between the talks, and gave also promising younger people the possibility to present their work, partly in more informal evening seminars. Even the contributions given on Friday afternoon and evening were attended by almost all participants, so we may assume that the program was greatly appreciated. A long standing question about algebraicity and transcendence of special values of Gauss hypergeometric functions has been answered in recent years using work of Wüstholz, Edixhoven, Yafaev and Klingler about conjectures concerning special subvarieties of Shimura varieties. The conference had a special day with four featured talks of these authors about the subject and an informal evening seminar concerning generalizations of this question to higher dimensions. A remarkable event on the classical number theoretic side of hypergeometric functions was Don Zagier's featured talk explaining a new link of complex differential equations to Hilbert and Teichmüller modular forms. The arithmetic-geometric mean can be written in terms of the hypergeometric function — a classical fact. Matsumoto–Shiga reported about possibilities of its genralization. Algebro-geometric aspects of the Painlevé equations and their generalizations were reported by van der Put and Tsuda. It is well known that these equations are related to the hypergeometric equations and its generalizations. About the classical hypergeometric equation, Vidunas proved that there are still many things to be studied, and Sasaki–Yoshida presented differential geometric and singularity theoretic aspects of the hyperbolic Schwarz map. The deep relation between hypergeometric differential equations and ball quotients originated with Schwarz, Picard, Terada, and Deligne–Mostow. New compactification techniques of such ball quotients (and more generally of so called Heegner divisor complements in such ball quotients) were reported by Heckman. These compactifications were introduced in full generality by Looijenga, inspired by the work of Heckman and Looijenga on the moduli space of rational elliptic surfaces. The geometric examples discussed in the lecture of Heckman were all related to groups generated by order four complex reflections (also called tetraflections) and go back to the work of Deligne–Mostow and of Kondo. Allcock presented a surprising conjecture about a ball quotient and the monster group. In this lecture the complex reflection groups were related to order three complex reflections (triflections in Conway's terminology), and in a sense the lecture by Heckman also served as a kind of introduction for Allcock's lecture. The conjecture by Allcock is truly amazing, and would in fact reveal a natural geometric surrounding (via generalized hypergeometric periods) for most of the sporadic groups. Holzapfel and his collaborators discussed towers of ball quotients and morphsims among these. A vast generalization of the hypergeometric equation developed during the last twenty years is the GKZ (Gelfand–Kapranov–Zelevinsky) equation was another important subject of the meeting. Beukers presented a criterion for their solvability by algebraic functions, and Alicia Dickenstein reported about examples admitting rational solutions.

UNIVERSAL NEAR-HORIZON CONFORMAL STRUCTURE AND BLACK HOLE ENTROPY

It is shown that a massless scalar probe reveals a universal near-horizon conformal structure for a wide class of black holes, including the BTZ. The central charge of the corresponding Virasoro algebra contains information about the black hole. With a suitable quantization condition on the central charge, the CFT associated with the black hole in our approach is consistent with the recent observation of Witten, where the dual theory for the BTZ in the AdS/CFT framework has been identified with the construction of Frenkel, Lepowsky and Meurman. This CFT admits the Fischer–Griess monster group as its symmetry. The logarithm of the dimension of a specific representation of the monster group has been identified by Witten as the entropy of the BTZ black hole. Our algebraic approach shows that a wide class of black holes share the same near-horizon conformal structure as that for the BTZ. With a suitable quantization condition, the CFT's for all these black holes in our formalism can be identified with the FLM model, although not through the AdS/CFT correspondence. The corresponding entropy for the BTZ provides a lower bound for the entropy of this entire class of black holes.

On the Action of the Sporadic Simple Baby Monster Group on its Conjugacy Class 2B

AbstractWe determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster groupBon its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 21+22.Co2. This is one of the first applications of a new general computational technique to enumerate big orbits.

Enumerating big orbits and an application: B acting on the cosets of [formula omitted

We describe a novel technique to handle big permutation domains for large groups. It is applied to the multiplicity-free action of the sporadic simple Baby Monster group on the cosets of its maximal subgroup Fi 23 , to determine the character table of the associated endomorphism ring.

Estimating the experimental value of the electromagnetic fine structure constant α¯0=1/137.036 using the Leech lattice in conjunction with the monster group and Spher’s kissing number in 24 dimensions

Abstract We start with various observations regarding the kissing number in the 24-dimensional Leech lattice N τ ( 24 ) = 196560 as well as the j-function coefficient b = 196884 and the minimal dimension in which the monster group can act Dm = 196883. Subsequently based on the previous results and earlier numerical experiments, we use a quibic potential to derive a quadratic equation x 2 + 128 20 x - ( N τ ( 24 ) / 10 ) = 0 where 128 = spin 7 = (2)7, 10 = D(10) and N τ ( 24 ) = 196560 are the spin representation, the super string dimension and the Leech kissing number, respectively. It is found that the only positive solution of this equation is x 1 = 137.036 = α ¯ 0 which is the accurate experimental value of inverse of the electromagnetic fine structure constant. This remarkable result is interpreted in terms of the connection between the Moonshine conjecture and string theory as well as the E-infinity relation between the kissing number in 10 dimensions K τ ( 10 ) = 336 and the degrees of freedom of Klein’s modular space dim Γ(7) = 336.

Constructing the Monster amalgam

The Monster group M contains a pair { C , N } of subgroups, where C ∼ 2 + 1 + 24 . Co 1 is the centralizer of an involution and N ∼ 2 2 + 11 + 2 ⋅ 11 . ( M 24 × Sym 3 ) is the normalizer of an elementary subgroup of order four with N ∩ C having index three in N. The amalgam { C , N } plays an important role in the existing constructions and uniqueness proofs for the Monster. We suggest a transparent construction of this amalgam.

Local conformal nets arising from framed vertex operator algebras

We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III 1 ) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel–Lepowsky–Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets.

An Elementary Approach to the Monster

is one of the most spectacular and mysterious mathematical achievements of the past fifty years. Unfortunately the various standard approaches to the Monster are notoriously difficult to learn. This paper presents a relatively elementary approach to the Monster. We describe a construction of the smallest non-Abelian finite simple group (of order 60), and show that it very closely parallels a construction of M x M. In section 2, before we begin our main discussion, we give a quick overview of the Monster both as a finite simple group and with respect to its most startling properties. This is done to allow those readers who have only a passing acquaintance with the Monster to get a better sense of its role in mathematics. Despite its brevity, this section has a vast scope, so we refer the reader to other sources for more comprehensive treatments (see [1], [2], or [6]). In section 3 we start with the tetrahedral graph and use Coxeter groups, along with certain additional relations, to obtain a group presentation for the finite simple group As. Then in section 4 we see that this method, when applied to different initial graphs, leads not only to the square M x M of the elusive Monster group, but to other finite simple groups as well.

Conjugacy Class Representatives in the Monster Group

AbstractThe paper describes a procedure for determining (up to algebraic conjugacy) the conjugacy class in which any element of the Monster lies, using computer constructions of representations of the Monster in characteristics 2 and 7. This procedure has been used to calculate explicit representatives for each conjugacy class.

On point orbits of the Co2-minimal parabolic geometry

The orbits of certain subgroups of Co 2, Conway's second largest simple group, on the points of Co 2's minimal parabolic geometry are determined. This information is used in [6,7] to analyse a grap associated with the Baby Monster group.

Generating triples of involutions of large sporadic groups

In each finite simple sporadic group, excepting the Baby Monster group B, the Monster group M, the McLaughlin group McL and Mathieu groups M11, M22 , M23, three generating involutions, two of which commute, are found.

Generalized Kac–Moody algebras, automorphic forms and Conway's group I

The moonshine properties imply that the twisted denominator identities coming from the action of the monster group on the monster algebra define modular forms. In this paper, we motivate the conjecture that the action of an extension of Conway's simple group Co 1 on the fake monster algebra gives rise to automorphic forms of singular weight on Grassmannians. We prove the conjecture for elements with square-free level and nontrivial fixpoint lattice.