Sporadic groups and string theory

whereW is the Weyl group, ρ is the Weyl vector, and the product is over all positive roots α. This is a special case of Weyl’s character formula which says that the character of a finite dimensional representation is equal to a sum similar to the left hand side divided by the product on the right hand side; for the 1-dimensional representation the character is 1 so the sum is equal to the product. The Weyl character formula looks rather like the other product formulas. This similarity was explained by Macdonald and Kac as follows. Macdonald observed that the Weyl denominator formula was just a statement about finite root systems, and generalized this to affine root systems, producing a collection of identities called the Macdonald identities. The Macdonald identity for the simplest affine root system is just the Jacobi triple product identity. Kac observed that the Macdonald identities were just the denominator formulas for the Kac-Moody algebras that he and Moody had discovered a few years before, and went on to prove a character formula for representations of these algebras generalizing the Weyl character formula. The Weyl-Kac denominator formula [12] for Kac-Moody algebras is