Rigidity For The Hopf Algebra Of Quasisymmetric Functions

We investigate the rigidities of the Hopf algebra of quasisymmetric functions with respect to monomial and fundamental basis respectively.We study some com-binatorial properties of composition posets arising from the analogous Pieri rules for quasisymmetric functions.As further applications,we obtain the following re-sults:?1?the linear map induced by sending M_? to M_?^r is the unique nontrivial graded algebra automorphism that takes the monomial basis into itself,but there are no nontrivial coalgebra automorphisms preserving the monomial basis;?2?the linear maps induced by sending F_? to F_?^c,F_?^r and F_?^t respectively are the only three graded algebra automorphisms preserving the fundamental basis.Moreover,the linear map induced by sending F_? to F_?^c is the unique nontrival graded coal-gebra automorphisms preserving the fundamental basis.Therefore,it is the only nontrivial graded Hopf algebra automorphism that takes the fundamental basis into itself.