Noncommutative Schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood–Richardson and Murnaghan–Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.