The question is in the title, however; there is a Hopf algebra of quasisymmetric functions which has the Hopf algebra of symmetric functions as a sub - Hopf algebra. The quasisymmetric functions have a basis of fundamental quasisymmetric functions. The quasisymmetric expansion of a symmetric function means writing it as a linear combination of fundamental quasisymmetric functions.
The inner product of two homogeneous symmetric functions is only nonzero if they have the same degree. Let $\chi$ and $\chi'$ be two characters of the symmetric group $S_r$. Let $\mathrm{ch}$ be the characteristic map. The inner product $\ast$ is characterised by $\mathrm{ch}(\chi)\ast\mathrm{ch}(\chi')=\mathrm{ch}(\chi\chi')$.
I am interested in the inner product of two Schur functions. This is a symmetric function and so is a linear combination of Schur functions. It is a known open problem to give a combinatorial interpretation of these coefficients.
I am asking a variation of this question, namely, is anything known about the quasisymmetric expansion of the inner product of two Schur functions?
