?WHAT IS... The Monster? Richard E. Borcherds When I was a graduate student, my supervisor John wishes to multiply elements of the monster ex- Conway would bring into the department his one- plicitly, R. A. Wilson can supply two matrices that year-old son, who was soon known as the baby generate the monster. But there is a catch: Each ma- monster. A more serious answer to the title ques- trix takes up about five gigabytes of storage, and tion is that the monster is the largest of the to quote from Wilson’s atlas page: “[S]tandard gen- (known1) sporadic simple groups. Its name comes erators have now been made as 196882 × 196882 from its size: The number of elements is matrices over GF(2)…. They have been multiplied together, using most of the computing resources 8080, 17424, 79451, 28758, 86459, 90496, of Lehrstuhl D für Mathematik, RWTH Aachen, for 17107, 57005, 75436, 80000, 00000 about 45 hours….” (The difficulty of multiplying two elements of the monster is caused not so much =246.320.59.76.112.133.17. by its huge size as by the lack of “small” repre- 19.23.29.31.41.47.59.71, sentations; for example, the symmetric group S50 about equal to the number of elementary particles is quite a lot bigger than the monster, but it only in the planet Jupiter. takes a few minutes to multiply two elements by The monster was originally predicted to exist by hand.) Finally the modular representations of the B. Fischer and by R. L. Griess in the early 1970s. Griess monster for large primes were worked out by constructed it a few years later in an extraordinary G. Hiss and K. Lux; the ones for small primes still tour de force as the group of linear transformations seem to be out of reach at the moment. on a vector space of dimension 196883 that preserve In the late 1970s John McKay decided to switch a certain commutative but nonassociative bilinear from finite group theory to Galois theory. One product, now called the Griess product. function that turns up in Galois theory is the elliptic Our knowledge of the structure and represen- modular function tations of the monster is now pretty good. The j(τ)=q−1 + 744 + 196884q + 21493760q2 + ··· 194 irreducible complex representations were n worked out by Fischer, D. Livingstone, and M. P. = c(n)q Thorne (before the monster was even shown to (q = e2πiτ), which is essentially the simplest non- exist). These take up eight large pages in the atlas constant function invariant under the action [A] of finite groups, which is the best single source τ → (aτ + b)/(cτ + d) of SL (Z) on the upper half of information about the monster (and other finite 2 plane {τ| (τ) > 0}. He noticed that the coefficient simple groups). The subgroup structure is mostly 196884 of q1 was almost equal to the degree known; in particular there is an almost complete 196883 of the smallest complex representation of list of the maximal subgroups, and the main gaps the monster (up to a small experimental error). in our knowledge concern embeddings of very The term “moonshine” roughly means weird rela- small simple groups in the monster. If anyone tions between sporadic groups and modular func- tions (and anything else) similar to this. It was Richard E. Borcherds is professor of mathematics at the clear to many people that this was just a mean- University of California, Berkeley. His work is supported in part by NSF grant DMS-9970611. His email address ingless coincidence; after all, if you have enough is [email protected], and his homepage is large integers from various areas of mathematics, http://www.math.berkeley.edu/~reb/. then a few are going to be close just by chance, and 1The announcement of the classification of the finite John McKay was told that his observation was simple groups about twenty years ago was a little over- about as useful as looking at tea leaves. John enthusiastic, but a recent 1,300-page preprint by Thompson took McKay’s observation further and M. Aschbacher and S. D. Smith should finally complete it. pointed out that the next few coefficients of the 1076 NOTICES OF THE AMS VOLUME 49, NUMBER 9 elliptic modular function were also simple linear that the symmetric group S3 is a group of diagram combinations of dimensions of irreducible repre- automorphisms of the Lie algebra D4. The monster sentations of the monster; for example, 21493760 Lie algebra has a denominator formula, similar to = 21296876 + 196883 + 1 . He suggested that there the Weyl denominator formula for finite-dimen- should be a natural infinite-dimensional graded sional Lie algebras and the Macdonald-Kac identi- representation V = n∈Z Vn of the monster such ties for affine Lie algebras, which looks like that the dimension of V is the coefficient c(n) of n − −1 − m n c(mn) qn in j(τ), at least for n =0 . (The constant term of j(σ ) j(τ)=p (1 p q ) m>0 j(τ) is arbitrary, since adding a constant to j still n∈Z produces a function invariant under SL2(Z) and is 2πiσ 2πiτ set equal to 744 mainly for historical reasons.) where p = e , q = e . This formula was dis- Conway and Norton [C-N] followed up Thompson’s covered independently in the 1980s by several peo- ple, including M. Koike, S. P. Norton, and D. Zagier. suggestion of looking at the McKay-Thompson se- There are similar identities with j(τ) replaced by ries T (τ)= Trace(g|V )qn , whose coefficients g n n the McKay-Thompson series of any element of the are given by the traces of elements g of the mon- monster, and C. J. Cummins and T. Gannon showed ster on the representations V , and found by cal- n that any function satisfying such identities is a culating the first few terms that these functions all Hauptmodul. So this provides some sort of expla- seemed to be Hauptmoduls of genus 0. (A Haupt- nation of Conway and Norton’s observation that the modul is a function similar to j but invariant under McKay-Thompson series are all Hauptmoduls. some group other than SL (Z).) A. O. L. Atkin, 2 So the question “What is the monster?” now has P. Fong, and S. D. Smith showed by computer cal- several reasonable answers: culation that there was indeed an infinite-dimen- 1. It is the largest sporadic simple group or al- sional graded representation of the monster whose ternatively the unique simple group of its order. McKay-Thompson series were the Hauptmoduls 2. It is the automorphism group of the Griess found by Conway and Norton, and soon afterwards algebra. I. B. Frenkel, J. Lepowsky, and A. Meurman explic- 3. It is the automorphism group of the monster itly constructed this representation using vertex vertex algebra. (This is probably the best an- operators. swer.) If a group acts on a vector space it is natural to 4. It is a group of diagram automorphisms of the ask if it preserves any algebraic structure, such as monster Lie algebra. a bilinear form or product. The monster module Unfortunately none of these definitions is constructed by Frenkel-Lepowsky-Meurman has a completely satisfactory. At the moment all con- vertex algebra structure invariant under the ac- structions of the algebraic structures above seem tion of the monster. Unfortunately there is no easy artificial; they are constructed as sums of two or way to explain what vertex algebras are; see [K] for more apparently unrelated spaces, and it takes a the best introduction to them. Roughly speaking, lot of effort to define the algebraic structure on the vertex algebras can be thought of (at least in char- sum of these spaces and to check that the monster acterstic 0) as commutative rings with derivation acts on the resulting structure. It is still an open where the ring multiplication is not quite defined problem to find a really simple and natural everywhere; this is analogous to rational maps in construction of the monster vertex algebra. algebraic geometry, which are also not quite defined everywhere. A more concrete but less intuitive de- References finition of a vertex algebra is that it consists of a [A] J. H. CONWAY, R. T. CURTIS, S. P. NORTON, R. A. PARKER, space with a countable number of bilinear products and R. A. WILSON, Atlas of Finite Groups, Clarendon satisfying certain rather complicated identities. In Press, Oxford, 1985; see also the online Atlas of Fi- the case of the monster vertex algebra V = Vn, nite Group Representations at http://www.mat. bham.ac.uk/atlas/v2.0/. this gives bilinear maps from Vi × Vj to Vk for all integers i,j,k, and the special case of the map [C-N] J. H. CONWAY and S. P. NORTON, Monstrous moonshine, Bull. London. Math. Soc. 11 (1979), 308–39. from V2 × V2 to V2 is (essentially) the Griess prod- [K] V. KAC, Vertex Algebras for Beginners, second edition, uct. So the Griess algebra is a sort of section of the University Lecture Series, vol. 10, American Mathe- monster vertex algebra. matical Society, Providence, RI, 1998. Following an idea of Frenkel, one can use the monster vertex algebra and the Goddard-Thorn The “WHAT IS...?” column carries short (one- or “no-ghost theorem” from string theory to construct two-page), nontechnical articles aimed at graduate stu- the monster Lie algebra. This is a Z2 -graded Lie dents. Each article focuses on a single mathematical ob- algebra whose piece of degree (m, n) ∈ Z2 has ject, rather than a whole theory.