The peak algebra in noncommuting variables

The well-known descent-to-peak map ΘQSym for the Hopf algebra of quasisymmetric functions, QSym, and the peak algebra $Π$ were originally defined by Stembridge in 1997. We introduce their noncommutative analogues, the labelled descent-to-peak map ΘNCQSym for the Hopf algebra of quasisymmetric functions in noncommuting variables, NCQSym, and the peak algebra in noncommuting variables NCΠ. Then, we define the Hopf algebra of Schur Q-functions in noncommuting variables. We show that our generalizations possess many properties analogous to their classical counterparts. Furthermore, we show that the coefficients in the expansion of certain elements of NCΠ in the monomial basis of NCQSym satisfy the generalized Dehn-Sommerville equation of Bayer and Billera. In the end, we give representation-theoretic interpretations of the descent-to-peak map for the Hopf algebras of symmetric functions and noncommutative symmetric functions.