The peak algebra in noncommuting variables

The well-known descent-to-peak map $\Theta_{\mathrm{QSym}}$ for the Hopf algebra of quasisymmetric functions, $\mathrm{QSym}$, and the peak algebra $\Pi$ were originally defined by Stembridge in 1997. We introduce their noncommutative analogues, the labelled descent-to-peak map $\Theta_{\mathrm{NCQSym}}$ for the Hopf algebra of quasisymmetric functions in noncommuting variables, $\mathrm{NCQSym}$, and the peak algebra in noncommuting variables $\mathrm{NC}\Pi$. 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 $\mathrm{NC}\Pi$ in the monomial basis of $\mathrm{NCQSym}$ 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.