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.
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Friedmann, Tamar and Stanley, Richard P., "Counting Conjugacy Classes of Elements of Finite Order in Lie Groups" (2014). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.