Rigidity for the Hopf algebra of quasi-symmetric functions

We investigate the rigidity of the Hopf algebra of quasi-symmetric functions with respect to monomial, fundamental and quasi-symmetric Schur basis respectively. It is showed that the linear map induced by sending $M_{\alpha}$ to $M_{\alpha^r}$ is the unique nontrivial graded algebra automorphism that takes the monomial basis into itself, but it does not preserve the comultiplication. We also show that the linear map induced by sending $F_{\alpha}$ to $F_{\alpha^c}$ is the unique nontrivial graded coalgebra automorphism preserving the fundamental basis. However, there are no nontrivial graded Hopf algebra automorphisms that preserve the quasi-symmetric Schur basis.