Mark Ronan's website
The Monster
|
|
The Monster group is the
largest of 26 exceptions discovered in the classification of all finite
'symmetry atoms', more properly known as 'finite simple groups'.
The rest of them fit into a well-understood table, a sort of periodic table,
but the exceptions — known as sporadic groups
— form a fascinating collection. All but six of them lie within the
Monster, though their discoveries were largely independent of it. The story
of discovery is described in Symmetry and
the Monster. The size of the Monster is 246.320.59.76.112.133.17.19.23.29.31.41.47.59.71,
which works out to be 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. The Monster is more than just
the largest exception — the Moonshine phenomena
connect it to number theory, and to string theory in mathematical physics.
These connections were entirely unsuspected when the Monster first emerged,
via one of its cross-sections, discovered by Bernd Fischer. This cross-section,
later called the Baby Monster, needs more than 4,000 dimensions in which to
operate, but the Monster itself needs 196,883 dimensions.
This number is the product of the three largest prime numbers that are
divisors of the Monster's size, namely 47, 59 and 71. The Monster was eventually
constructed by Robert Griess as the
symmetry group of an algebra structure in 196,884 dimensions.
His work split the space into three subspaces, and his main task was to show
there were symmetries intermingling these subspaces. The dimensions of the
subspaces are: 98,304 + 300 +
98,280 = 196,884 The first number 98,304 = 212 × 24
comes from the Golay code in 24 dimensions. The second number 300 = 24 + 23 + 22 + .
. . + 3 + 2 + 1 is the dimension of the space
of 24‑by‑24 symmetric matrices. The third number 96,280 = 196,560 ÷ 2
comes from the Leech Lattice in 24 dimensions,
where there are 196,560 vertices closest to a given vertex, forming 98,280
diametrically opposite pairs. The Monster's Character
Table The finite
symmetry atoms are very large, and data about each one is encoded into a character table — a square array of
numbers, rather like a giant sudoku puzzle. The Monster's table has 194 rows
and columns, and the Moonshine connections showed
that the first column generates an important sequence of numbers in number
theory (the coefficients of the j‑function).
Other columns can be used in a similar way, and these moonshine connections
eventually created a link to the mathematical physics of string theory. There are still mysteries
associated with the Monster. Here is one. The 194 columns of the Monster's
character table span a space of 163 dimensions. The number 163 is well-known
in number theory because the square root of -163 yields an extension of the
rational numbers having unique factorisation, and 163 is by
far the largest integer having this property. |
|