Symmetry and the Monster

One of the greatest quests in mathematics

A little early history

Equations of degree 2—meaning the highest power of x is x2: solved by the Babylonians in about 1800 BC

Equations of degree 3: solved using a graphical method by Omar Khayyám in about 1100 AD

Equations of degrees 3 and 4: solved by Italian mathematicians in the first half of the 1500s.

The quintic equation

• Equations of degree 5 were a problem. No one could come up with a formula.

• 1799 Paolo Ruffini

• 1824 Niels Hendrik Abel

• Early 1830s Évariste Galois

Galois’s Ideas

• If the equation is irreducible any solution is equivalent to any other.

• The solutions can be permuted among one another.

• Not all permutations are possible, but those that are form the Galois group of the equation.

x4 - 10x2 + 1 = 0

• There are four solutions, a, b, c, d

• The negative of a solution is a solution, so we can set: a + b = 0, and c + d = 0.

• This restricts the possible permutations;if a goes to b then b goes to a, if a goes to c then b goes to d.

The Galois Group

• Galois investigated when the solutions to a given equation can be expressed in terms of roots, and when they can’t.

• The solutions can be deconstructed into roots precisely when the Galois group can be deconstructed into cyclic groups.

Atoms of Symmetry

• A group that cannot be deconstructed into simpler groups is called simple.

• For each prime number p the group of rotations of a regular p-gon is simple; it is a cyclic group.

• The structure of a non-cyclic simple group can be very complex.

Families of Simple Groups

• Galois discovered the first family of non-cyclic finite simple groups.

• Other families were discovered in the later nineteenth century.

• All these families were later seen as ‘groups of Lie type’, stemming from work of Sophus Lie.

Sophus Lie

• Lie wanted to do for differential equations what Galois had done for algebraic equations.

• He created the concept of continuous groups, now called Lie groups.

• ‘Simple’ Lie groups were classified into seven families, A to G, by Wilhelm Killing.

Finite groups of Lie type

• Finite versions of Lie groups are called groups of Lie type.

• Most of them were created by Leonard Dickson in 1901.

• In 1955 Claude Chevalley found a uniform method yielding all families A to G.

• Variations on Chevalley’s theme soon emerged, and by 1961 all finite groups of Lie type had been found.

The Feit-Thompson Theorem

• In 1963, Walter Feit and John Thompson proved the following big theorem:

• A non-cyclic finite simple group must contain an element of order 2.

• Elements of order 2 give rise to ‘cross-sections’, and Richard Brauer had shown that knowing one cross-section of a finite simple group gave a firm handle on the group itself.

The Classification

• By 1965 it looked as if a finite simple group must be a group of Lie type, or one of five exceptions discovered in the mid-nineteenth century.

• These five exceptions, the Mathieu groups—created by Émile Mathieu—are very exceptional. There is nothing else quite like them.

A Cat among the Pigeons

• In 1966, Zvonimir Janko in Australia produced a sixth exception.

• He discovered it via one of its cross-sections.

• This led Janko and others to search for more exceptions, and within ten years another twenty turned up.

The Exceptions

• Some were found using the cross-section method

• Some were found by studying groups of permutations

• Some were found using geometry

The Hall-Janko group J2

• Janko found it using the cross-section method.

• Marshall Hall found it using permutation groups.

• Jacques Tits constructed it using geometry.

The Leech Lattice

• John Leech used the largest Mathieu group M24 to create a remarkable lattice in 24 dimensions.

• John Conway studied Leech’s lattice and turned up three new exceptions.

• Had he investigated it two years earlier, he would have found two more—the Leech Lattice contains half of the exceptional symmetry atoms.

Fischer’s Monsters

• Bernd Fischer in Germany discovered three intriguing and very large permutation groups, modelled on the three largest Mathieu groups.

• He then found a fourth one of a different type, and even larger, called the Baby Monster.

• Using this as a cross-section, he turned up something even bigger, called the Monster.

Computer Constructions

• When the exceptional groups were ‘discovered’, it was not always clear that they existed.

• Proving existence could be tricky, and computers were sometimes used.

• For example the Baby Monster was constructed on a computer.

• BUT the Monster was too large for computer methods.

Constructing the Monster

• Fischer, Livingstone and Thorne constructed the character table of the Monster, a 194-by-194 array of numbers.

• This showed the Monster could not live in fewer than 196,883 dimensions.

• 196,883 = 475971, the three largest primes dividing the size of the Monster.

• Later Robert Griess constructed the Monster by hand in 196,884 dimensions.

McKay’s Observation

• 196,883 + 1 = 196,884, the smallest non-trivial coefficient of the j-function.

• McKay wrote to Thompson who had further data on the Monster available.

• Thompson confirmed that other dimensions for the Monster seemed to be related to coefficients of the j-function.

Ogg’s Observation

• Shortly after evidence for the Monster was announced, Andrew Ogg attended a lecture in Paris.

• Jacques Tits wrote down the size of the Monster, as a product of prime numbers.

• Ogg noticed these were precisely the primes that appeared in connection with his own work on the j-function.

Moonshine

• The mysterious connections between the Monster and the j-function were dubbed Moonshine.

• John Conway and Simon Norton investigated them in detail, proved they were real, and made conjectures about a deeper connection.

• Their paper was called Monstrous Moonshine

Vertex Algebras and String Theory

• The Moonshine connections involved the Monster acting in finite dimensional spaces.

• Frenkel, Leopwski and Meurman combined these in an infinite dimensional space.

• Their space had a vertex algebra structure, which brought in the mathematics of string theory.

Conway-Norton Conjectures

• The conjectures by Conway and Norton were later proved by Richard Borcherds, who received a Fields Medal for his work,but as he points out, there are still mysteries to resolve

• For example the space of ‘j-functions’ associated with the Monster has dimension 163. Is this just a coincidence?

• e√163 = 262537412640768743.99999999999925... is very close to being a whole number.