Quasisymmetric functions and Heisenberg doubles

The ring of quasisymmetric functions is free over the ring of symmetric functions. This result was previously proved by M. Hazewinkel combinatorially through constructing a polynomial basis for quasisymmetric functions. The recent work by A. Savage and O. Yacobi on representation theory provides a new proof to this result. In this paper, we proved that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double. Examples satisfying the above conditions are discussed.