  In a previous Letter (J. Phys. A v.47 (2014) 212003) we have presented
numerical evidence that a Hamiltonian expressed in terms of the generators of
the periodic Temperley-Lieb algebra has, in the finite-size scaling limit, a
spectrum given by representations of the Virasoro algebra with complex highest
weights. This Hamiltonian defines a stochastic process with a Z_N symmetry. We
give here analytical expressions for the partition functions for this system
which confirm the numerics. For N even, the Hamiltonian has a symmetry which
makes the spectrum doubly degenerate leading to two independent stochastic
processes. The existence of a complex spectrum leads to an oscillating approach
to the stationary state. This phenomenon is illustrated by an example.
