Vertex operator algebra and McKay’s E8 observation on the Monster

As one of mysterious properties of the Monster simple group, J. McKay observed that there exists an interesting connection between the 2A-conjugacy class of the Monster and the (extended) E8 Dynkin diagram. Monster, the largest sporadic finite simple group, can be defined by the symmetry group of the moonshine vertex operator algebra. In this talk we will report on results of our research to find the structure of E8 diagram inside the moonshine vertex operator algebra. This is a joint work with Ching Hung Lam and Hiromichi Yamada. 1 McKay’s E8-observation on the Monster After the classification of finite simple groups, we found the Monster as the largest sporadic finite simple groupM. The MonsterM has many mysterious properties; probably the most famous example is known to as the moonshine phenomenon or the monstrous moonshine [CN, B]. Another example is McKay’s E8-observation on the Monster, which is the main topic of this talk. Let us explain McKay’s observation briefly. The Monster has two conjugacy classes of involutions, called 2A and 2B-conjugacy classes [ATLAS]. Let g, h ∈ M be two involutions of 2A-conjugacy class. Then the order of a product gh is known to be less than or equal to 6, which is sometimes referred to as the 6-transposition property, and the conjugacy class of the product falls into one of the following [ATLAS, C]: 1A, 2A, 3A, 4A, 5A, 6A, 4B, 2B, or 3C, where the name pX is used in the way that p denotes the order and the letter X is labeled to distinguish classes having the same order (cf. [ATLAS]). In [Mc], J. McKay pointed 1 out that the numbers above can be attached to the extended affine E8 diagram as its labels in the following way