The theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one. The resulting expressions, dubbed superpolynomials, provide the natural N = 1 supersymmetric version of the classical bases of symmetric functions. Here we consider the case where more than one independent anticommuting variable is attached to each ordinary variable. First, the N = 2 super-version of the monomial, elementary, homogeneous symmetric functions, as well as the power sums, are constructed systematically (using an exterior-differential formalism for the multiplicative bases), these functions being now indexed by a novel type of superpartitions. Moreover, the scalar product of power sums turns out to have a natural N = 2 generalization which preserves the duality between the monomial and homogeneous bases. All these results are then generalized to an arbitrary value of N . Finally, for N = 2, the scalar product and the homogeneous functions are shown to have a one-parameter deformation, a result that prepares the ground for the yet-to-be-defined N = 2 Jack superpolynomials.

The explicit formula for the superconformal singular vectors in the Neveu-Schwarz sector has been obtained recently, via its symmetric polynomial representation, as a sum of Jack superpolynomials. Here we present the analogous, but slightly more complicated, closed-form expression for the Ramond singular vectors.

We present a review of the most important results in the theory of symmetric functions in superspace (or symmetric superpolynomials), summarizing all principal contributions since its introduction in 2001 in the context of the supersymmetric Calogero-Moser-Sutherland integrable model. We also mention some open problems which remain unanswered at this moment, in particular the connection with representation theory. In addition, we provide a free open access source code, relying on SageMath library, that can be used as a research tool for symmetric superpolynomials. The content is directed to an audience new to this research area, but who is familiar with the classical theory of symmetric functions.

The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.

In a recent work, we have initiated the theory of N = 2 symmetric superpolynomials. As far as the classical bases are concerned, this is a rather straightforward generalization of the N = 1 case. However this construction could not be generalized to the formulation of Jack superpolynomials. The origin of this obstruction is unraveled here, opening the path for building the desired Jack extension. Those are shown to be obtained from the non-symmetric Jack polynomials by a suitable symmetrization procedure and an appropriate dressing by the anticommuting variables. This construction is substantiated by the characterization of the N = 2 Jack superpolynomials as the eigenfunctions of the N = 2 supersymmetric version of the Calogero-Sutherland model, for which, as a side result, we demonstrate the complete integrability by displaying the explicit form of four towers of mutually commuting (bosonic) conserved quantities. The N = 2 Jack superpolynomials are orthogonal with respect to the analytical scalar product (induced by the quantum-mechanical formulation) as well as a new combinatorial scalar product defined on a suitable deformation of the power-sum basis.