The University of Michigan Combinatorics Seminar
In 1972 Macdonald generalized the Weyl denominator identity to affine root systems. The simplest example of these identities turned out to be the famous Jacobi triple product identity. In 1994 V. G. Kac and M. Wakimoto conjectured an analog for affine Lie superalgebras and showed that it has applications in number theory. In this lecture we discuss the progress done on these problems. A proof for the exceptional affine case D(2,1,a) will be given. From this case we conclude a formula for counting the number of presentations of an integer as a sum of 8 squares.