Unimodality of Eulerian quasisymmetric functions

We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Q λ , j is Schur-positive, and moreover that the sequence Q λ , j as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type ( q , p ) -Eulerian polynomial A λ maj , des , exc ( q , p , q − 1 t ) is t-unimodal.