Symmetric functions in noncommuting variables in superspace

In 2004, Rosas and Sagan developed the theory of symmetric functions in noncommuting variables, achieving results analogous to classical symmetric functions. On the other hand, the same year, Desrosiers, Lapointe and Mathieu introduced the theory of symmetric functions in superspace, involving both commuting and anticommuting variables, extending the classic theory. Here, we introduce symmetric functions in noncommuting variables in superspace. We extend the classical symmetric functions in noncommuting variables to superspace: monomial, power sum, elementary and complete homogeneous functions. These functions generalize both those studied by Rosas and Sagan and those studied by Desrosiers, Lapointe, and Mathieu. Additionally, we define Schur–type functions in noncommuting variables in superspace.