An integral -dimensional lattice is the set of all integral linear combinations
of base vectors of , equipped with the usual (positive definite) inner product, satisfying
But then, is contained in its dual lattice , and if we say that is unimodular.
If all , we say that is an even lattice. Even unimodular lattices (such as the -lattice or the Niemeier lattices) are wonderful objects, but they can only live in dimensions which are multiples of .
Just like the Conway group is the group of rotations of the Leech lattice, one might ask whether there is a very special lattice on which the Monster group acts faithfully by rotations. If such a lattice exists, it must live in dimension at least .
Conway writes that Simon Norton showed ‘by a very simple computations that does not even require knowledge of the conjugacy classes, that any -dimensional representation of the Monster must support an invariant algebra’, which, after adding an identity element , we now know as the -dimensional Griess algebra.
Further, on page 529, Conway writes:
Norton has shown that the lattice spanned by vectors of the form , where and are transposition vectors, is closed under the algebra multiplication and integral with respect to the doubled inner product . The dual quotient is cyclic of order some power of , and we believe that in fact is unimodular.
Here, transposition vectors correspond to transpositions in , that is, elements of conjugacy class .
In his post, Adam considers the -dimensional lattice (which has as its rotation symmetry group), and asks for the minimal norm (squared) of a lattice point, which he believes is , and for the number of minimal vectors in the lattice, which might be
the number of oriented arcs in the Monster graph.
Here, the Monster graph has as its vertices the elements of in conjugacy class (which has elements) and with an edge between two vertices if their product in again belongs to class , so the valency of the graph must be , as explained in that old post the monster graph and McKay’s observation.
When I asked Adam whether he had more information about his lattice, he kindly informed me that Borcherds told him that the Norton lattice didn’t turn out to be unimodular after all, but that a unimodular lattice with monstrous symmetry had been constructed by Scott Carnahan in the paper A Self-Dual Integral Form of the Moonshine Module.
The major steps (or better, the little bit of it I could grasp in this short time) in the construction of this unimodular -dimensional monstrous lattice might put a smile on your face if you are an affine scheme aficionado.
Already in his paper Vertex algebras, Kac-Moody algebras, and the Monster, Richard Borcherds described an integral form of any lattice vertex algebra. We’ll be interested in the lattice vertex algebra constructed from the Leech lattice and call its integral form .
One starts with a fixed point free rotation of in of prime order , which one can lift to an automorphism of the vertex algebra of order giving an isomorphism of vertex operator algebras over .
For two distinct primes if has an element of order one can find one such such that and , and one can lift to an automorphism of such that as vertex operator algebras over .
Problem is that these lifts of automorphisms and the isomorphisms are not compatible with the integral form of , but ‘essentially’, they can be performed on
where is a primitive -th root of unity. These then give a -form on .
Next, one uses a lot of subgroup information about to prove that these -forms of have as their automorphism group.
Then, using all his for different triples in one can glue and use faithfully flat descent to get an integral form of the moonshine module with monstrous symmetry and such that the inner product on is positive definite.
Finally, one looks at the weight subspace of which gives us ourCarnahan’s-dimensional unimodular lattice with monstrous symmetry!
Beautiful as this is, I guess it will be a heck of a project to deduce even the simplest of facts about this wonderful lattice from running through this construction.
For example, what is the minimal length of vectors? What is the number of minimal length vectors? And so on. All info you might have is very welcome.
Whenever I visit someone’s YouTube or Twitter profile page, I hope to see an interesting banner image. Here’s the one from Richard Borcherds’ YouTube Channel.
Not too surprisingly for Borcherds, almost all of these numbers are related to the monster group or its moonshine.
Let’s try to decode them, in no particular order.
196884
John McKay’s observation was the start of the whole ‘monstrous moonshine’ industry. Here, and are the dimensions of the two smallest irreducible representations of the monster simple group, and is the first non-trivial coefficient in Klein’s j-function in number theory.
is also the dimension of the space in which Robert Griess constructed the Monster, following Simon Norton’s lead that there should be an algebra structure on the monster-representation of that dimension. This algebra is now known as the Griess algebra.
1729 is the second (and most famous) taxicab number. A long time ago I did write a post about the classic Ramanujan-Hardy story the taxicab curve (note to self: try to tidy up the layout of some old posts!).
“We’ve found that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named. It turns out that Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics.”
There’s no other number like responsible for the existence of sporadic simple groups.
24 is the length of the binary Golay code, with isomorphism group the sporadic Mathieu group and hence all of the other Mathieu-groups as subgroups.
24 is the dimension of the Leech lattice, with isomorphism group the Conway group (dotto), giving us modulo its center the sporadic group and the other Conway groups, and all other sporadics of the second generation in the happy family as subquotients (McL,HS,Suz and )
24 is the central charge of the Monster vertex algebra constructed by Frenkel, Lepowski and Meurman. Most experts believe that the Monster’s reason of existence is that it is the symmetry group of this vertex algebra. John Conway was one among few others hoping for a nicer explanation, as he said in this interview with Alex Ryba.
60 is, of course, the order of the smallest non-Abelian simple group, , the rotation symmetry group of the icosahedron. is the symmetry group of choice for most viruses but not the Corona-virus.
3264
3264 is the correct solution to Steiner’s conic problem asking for the number of conics in tangent to five given conics in general position.
Steiner himself claimed that there were such conics, but realised later that he was wrong. The correct number was first given by Ernest de Jonquières in 1859, but a rigorous proof had to await the advent of modern intersection theory.
Eisenbud and Harris wrote a book on intersection theory in algebraic geometry, freely available online: 3264 and all that.
248
248 is the dimension of the exceptional simple Lie group . is also connected to the monster group.
If you take two Fischer involutions in the monster (elements of conjugacy class 2A) and multiply them, the resulting element surprisingly belongs to one of just 9 conjugacy classes:
1A,2A,2B,3A,3C,4A,4B,5A or 6A
The orders of these elements are exactly the dimensions of the fundamental root for the extended Dynkin diagram.
163 is a remarkable number because of the ‘modular miracle’
This is somewhat related to moonshine, or at least to Klein’s j-function, which by a result of Kronecker’s detects the classnumber of imaginary quadratic fields and produces integers if the classnumber is one (as is the case for ).
The details are in the post the miracle of 163, or in the paper by John Stillwell, Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.
His description of the -function (at 4:13 in the movie) is simply hilarious!
Borcherds connects to the monster moonshine via the -function, but there’s another one.
The monster group has conjugacy classes and monstrous moonshine assigns a ‘moonshine function’ to each conjugacy class (the -function is assigned to the identity element). However, these functions are not linearly independent and the space spanned by them has dimension exactly .
SNORT, invented by Simon NORTon is a map-coloring game, similar to COL. Only, this time, neighbours may not be coloured differently.
SNORTgo, similar to COLgo, is SNORT played with go-stones on a go-board. That is, adjacent stones must have the same colour.
SNORT is a ‘hot’ game, meaning that each player is eager to move as most moves will improve your position. In COL players are reluctant to move, because a move limits your next moves.
For this reason, SNORT positions are much harder to evaluate, and one needs the full force of Conways’s ONAG.
Here’s a SNORTgo endgame. Who has a winning strategy?, and what is the first move in that strategy?
The method to approach such an endgame is similar to that in COLgo. First we remove all dead spots from the board.
What remains, are a 4 spots available only to Right (white) and 5 spots available only to Left (bLack). Further, there a 3 ‘live’ regions: the upper righthand corner and the two lower corners.
The value of these corners must be computed inductively.
Here’s the answer:
For example, Right’s best option in the left-most game (corresponding to the upper righthand corner of the endgame) is to put his stone on N12, resulting in a game in which neither player can move (the zero game).
On the other hand, Left can put a stone at either N11, N12 or N13 leaving a game in which she has two more moves, whereas Right han none (the game).
The other positions are computed similarly.
To get the value of the endgame we have to sum up all these values.
This can either be done using the addition rule given in ONAG, or by using programs in combinatorial game theory.
There’s Combinatorial Game Suite, developed by Aaron Siegel. But, for some reason I can no longer use it on macOS High Sierra.
Fortunately, the older program David Wolfe’s toolkit is still available, and runs on my MacBook.
The sum game evaluates to , which is a ‘fuzzy’ game, that is, its value is confused with .
This means that the first player to move has a winning strategy in the endgame.
Can you spot the (unique) winning move for Right (white) and one (of two) winning move for Left (bLack)?