Symmetric functions in noncommuting variables in superspace

Abstract: 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, in 2004, Desrosiers, Lapointe and Mathieu introduced the theory of symmetric functions in superspace, which involve both commuting and anticommuting variables, extending the classic theory. Here, we introduce symmetric functions in noncommuting variables in superspace. We define the classical symmetric functions in noncommuting variables to superspace: monomials, power sums, elementaries and complete homogeneous, which generalize both the ones studied by Rosas and Sagan and the ones studied by Desrosiers, Lapointe and Mathieu. We also define Schur--type functions in noncommuting variables in superspace.