Plethystic exponential calculus and characteristic polynomials of permutations

Abstract

We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of symmetric products of real/algebraic tori. The proof uses formal power series expansions of plethystic exponentials, and has been motivated by some recent applications of these combinatorial tools in supersymmetric gauge and string theories. Since the methods are elementary, we tried to be self-contained, and relate to other topics such as the q-binoomial theorem, and the cycle index and Molien series for the symmetric group.