• DOI:10.5040/9781474210379.ch-010
• Corpus ID: 118247695

What is the monster

@article{Borcherds2002WhatIT,
  title={What is the monster},
  author={Richard E. Borcherds},
  journal={arXiv: Group Theory},
  year={2002}
}

• R. Borcherds
• Published 24 September 2002
• Mathematics
• arXiv: Group Theory

about equal to the number of elementary particles in the planet Jupiter. The monster was originally predicted to exist by Fischer and by Griess in the early 1970’s. Griess constructed it a few years later in an extraordinary tour de force as the group of linear transformations on a vector space of dimension 196883 that preserved a certain commutative but non-associative bilinear product, now called the Griess product. Our knowledge of the structure and representations of the monster is now pretty good. The 194 irreducible complex representations were worked out by Fischer, Livingstone and Thorne (before the monster was even shown to exist) They take up 8 large pages in the atlas [A] of finite groups, which is the best single source of information about the monster (and other finite simple groups). The subgroup structure is mostly known; in particular there is an almost complete list of the maximal subgroups, and the main gaps in our knowledge concern embeddings of very small simple groups in the monster. If anyone wishes to multiply elements of the monster explicitly, R. A. Wilson can supply two matrices that generate the monster. But there is a catch: each matrix takes up about 5 gigabytes of storage, and to quote from Wilson’s atlas page: “standard generators have now been made as 196882×196882 matrices over GF(2)... They have been multiplied together, using most of the computing resources of Lehrstuhl D für Mathematik, RWTH Aachen for about 45 hours...”. (The difficulty of multiplying two elements of the monster is caused not so much by its huge size as by the lack of “small” representations; for example, the symmetric group S50 is quite a lot bigger than the monster, but it only takes a few minutes to multiply two elements by hand.) Finally the modular representations of the monster for large primes were worked out by Hiss and Lux; the ones for small primes still seem to be out of reach at the moment.