  We determine the Grothendieck ring of finite-dimensional comodules for the
free Hopf algebra on a matrix coalgebra, and similarly for the free Hopf
algebra with bijective antipode and other related universal quantum groups. The
results turn out to be parallel to those for Wang and Van Daele's deformed
universal compact quantum groups and Bichon's generalization of those results
to universal cosovereign Hopf algebras: in all cases the rings are isomorphic
to those of non-commutative polynomials over certain sets, these sets varying
from case to case. In most cases we are able to give more precise information
about the multiplication table of the Grothendieck ring.
