Quasisymmetric functions and Heisenberg doubles

Abstract

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