Vertex Representations via Finite Groups and the Mckay Correspondence

where the first factor is a symmetric algebra and the second one is a group algebra. The affine algebra ĝ contains a Heisenberg algebra ĥ. One can define the so-called vertex operators X(α, z) associated to α ∈ Q acting on V essentially using the Heisenberg algebra ĥ. The representation of ĝ on V is then obtained from the action of the Heisenberg algebra ĥ and the vertex operators X(α, z) associated to α in the root system of g. This construction was extended in [F2] to more general lattices to provide vertex representations of affinization of Kac-Moody Lie algebras. The special case of affinization of affine Lie algebras called toroidal Lie algebras was discussed further in [MRY]. Another important special case of vertex representations is given by the standard lattice Z . The vertex operators corresponding to the unit vectors in Z give rise to a representation of an infinitedimensional Clifford algebra [F1], the relation known as boson-fermion correspondence. In the special case N = 1, the transition matrix between the monomial bases for representations of Heisenberg and Clifford algebras yields the character tables of symmetric groups Sn for all n [F3] (see [J1]).