Reduced unitary matrix models and the hierarchy of τ-functions

We study reductions of unitary one-matrix models. The unitary model admits an especially rich class of reductions of which the widely known symmetric model is only one. The partition function of a certain class of reduced models is shown to be a product of distinct Toda chain τ-functions. Virasoro constraints are also derived for the case of unitary models. It is claimed, in analogy with the reduced hermitian model, that in the continuous limit the Virasoro constraints must be imposed on a fractional power of the original partition function.