  Using the symmetric group $S_Q$ symmetry of the $Q$-state Potts model, we
classify the (scalar) operator content of its underlying field theory in
arbitrary dimension. In addition to the usual identity, energy and
magnetization operators, we find fields that generalize the $N$-cluster
operators well-known in two dimensions, together with their subleading
counterparts. We give the explicit form of all these operators -- up to
non-universal constants -- both on the lattice and in the continuum limit for
the Landau theory. We compute exactly their two- and three-point correlation
functions on an arbitrary graph in terms of simple probabilities, and give the
general form of these correlation functions in the continuum limit at the
critical point. Specializing to integer values of the parameter $Q$, we argue
that the analytic continuation of the $S_Q$ symmetry yields logarithmic
correlations at the critical point in arbitrary dimension, thus implying a
mixing of some scaling fields by the scale transformation generator. All these
logarithmic correlation functions are given a clear geometrical meaning, which
can be checked in numerical simulations. Several physical examples are
discussed, including bond percolation, spanning trees and forests, resistor
networks and the Ising model. We also briefly address the generalization of our
approach to the $O(n)$ model.
