Gauged permutation invariant matrix quantum mechanics: path integrals

We give a path integral construction of the quantum mechanical partition function for gauged finite groups. Our construction gives the quantization of a system of d, N × N matrices invariant under the adjoint action of the symmetric group S_N. The approach is general to any discrete group. For a system of harmonic oscillators, i.e. for the non-interacting case, the partition function is given by the Molien-Weyl formula times the zero-point energy contribution. We further generalise the result to a system of non-square and complex matrices transforming under arbitrary representations of the gauge group.