Counting conjugacy classes of elements of finite order in Lie groups

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define N ( G , m ) to be the number of conjugacy classes of elements of finite order m in a Lie group G , and N ( G , m , s ) to be the number of such classes whose elements have s distinct eigenvalues or conjugate pairs of eigenvalues. What is N ( G , m ) for G a unitary, orthogonal, or symplectic group? What is N ( G , m , s ) for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques. It appears that the second question has not been asked before; here it is inspired by questions related to enumeration of vacua in string theory. Our combinatorial methods allow us to answer both questions.