  In a previous paper I have defined a new basis for the representation ring of
a Weyl group. In this paper we show that the new basis is related to the
standard basis by an upper triangular unipotent matrix. We also give a new
parametrization of representations in a fixed family by certain pairs of
subgroups of a finite group attached to the family. We outline an extension to
the case of unipotent representations of a finite Chevalley group.
