We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric Macdonald polynomials by $t$-symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of $R_m$. We define $m$-symmetric Schur functions through a somewhat complicated process involving their dual basis, tableaux combinatorics, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the $m$-symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of $m$-symmetric Schur functions. We obtain relations on the $(q,t)$-Koska coefficients $K_{\Omega \Lambda}(q,t)$ in the $m$-symmetric world, and show in particular that the usual $(q,t)$-Koska coefficients are special cases of the $K_{\Omega \Lambda}(q,t)$'s. Finally, we show that when $m$ is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials.

The fundamental quasisymmetric functions in superspace are a generalization of the fundamental quasisymmetric functions involving anticommuting variables. We obtain the action of the product, coproduct, and antipode on the fundamental quasisymmetric functions in superspace. We also extend to superspace the well known expansion of the Schur functions in terms of fundamental quasisymmetric functions.

Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and $t$-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric Macdonald polynomials satisfies a symmetry property generalizing that satisfied by the usual Macdonald polynomials. We then obtain Pieri rules for the bisymmetric Macdonald polynomials where the sums are over certain vertical strips.

The m-symmetric Macdonald polynomials form a basis of the space of polynomials that are symmetric in the variables x_m+1,x_m+2,… (while having no special symmetry in the variables x_1,…,x_m).We establish in this article the fundamental properties of the m-symmetric Macdonald polynomials. These include among other things the orthogonality with respect to a natural scalar product, as well as formulas for the squared norm, the evaluation, and the inclusion. We also obtain a Cauchy-type identity for the m-symmetric Macdonald polynomials which specializes to the known Cauchy-type identity for the non-symmetric Macdonald polynomials.

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.

We demonstrate the validity of previously conjectured explicit expressions for the norm and the evaluation of the Macdonald polynomials in superspace. These expressions, which involve the arm-lengths and leg-lengths of the cells in certain Young diagrams, specialize to the well known formulas for the norm and the evaluation of the usual Macdonald polynomials.

We introduce Macdonald polynomials indexed by $n$-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the $q,t$-Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to $q,t$-analogs of the dimensions of the irreducible representations of $C_n \sim S_d$, the wreath product of the cyclic group $C_n$ with the symmetric group.

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.

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in α (expressed, as in the usual Jack polynomial case, in terms of certain hook lengths in a Ferrers’ diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.

Generalizations of the (rank- 1 1 ) Bannai–Ito algebra are obtained from a refinement of the grade involution of the Lie superalgebra o s p ( 1 , 2 ) \mathfrak {osp}(1,2) . A hyperoctahedral extension is derived by using a realization of o s p ( 1 , 2 ) \mathfrak {osp}(1,2) in terms of Dunkl operators associated with the Weyl group B 3 B_3 .

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.

Generalizations of the (rank 1) Bannai-Ito algebra are obtained from a refinement of the grade involution of the Lie super algebra $\mathfrak{osp}(1,2)$. A hyperoctahedral extension is derived by using a realization of $\mathfrak{osp}(1,2)$ in terms of Dunkl operators associated to the Weyl group $B_3$.

The Schur functions in superspace sΛ and s¯Λ are the limits q=t=0 and q=t=∞ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases sΛ and s¯Λ (which happen to be essentially dual). As a consequence, we derive the basic properties of these bases such as dualities, monomial expansions, and tableaux generating functions.

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 two independent anticommuting variables are attached to each ordinary variable. The N=2 super-version of the monomial, elementary, homogeneous symmetric functions, as well as the power sums, are then 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 homogenous 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.

Generalizations of the (rank 1) Bannai-Ito algebra are obtained from a refinement of the grade involution of the Lie super algebra $\mathfrak{osp}(1,2)$. A hyperoctahedral extension is derived by using a realization of $\mathfrak{osp}(1,2)$ in terms of Dunkl operators associated to the Weyl group $B_3$.

Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials. The starting point of this work is the observation of a previously unnoticed stability property of the Macdonald superpolynomials when the fermionic sector m is sufficiently large: their decomposition in the monomial basis is then independent of m . These stable superpolynomials are readily mapped into bisymmetric polynomials, an operation that spoils the ring structure but drastically simplifies the associated vector space. Our main result is a factorization of the (stable) bisymmetric Macdonald polynomials, called double Macdonald polynomials and indexed by pairs of partitions, into a product of Macdonald polynomials (albeit subject to non-trivial plethystic transformations). As an off-shoot, we note that, after multiplication by a t -Vandermonde determinant, this provides explicit formulas for a large class of Macdonald polynomials with prescribed symmetry. The factorization of the double Macdonald polynomials leads immediately to the generalization of basically every elementary properties of the Macdonald polynomials to the double case (norm, kernel, duality, evaluation, positivity, etc). When lifted back to superspace, this validates various previously formulated conjectures in the stable regime. The q,t -Kostka coefficients associated to the double Macdonald polynomials are shown to be q,t -analogs of the dimensions of the irreducible representations of the hyperoctahedral group B_n . Moreover, a Nabla operator on the double Macdonald polynomials is defined, and its action on a certain bisymmetric Schur function can be interpreted as the Frobenius series of a bigraded module of dimension (2n+1)^n , a formula again characteristic of the Coxeter group of type B_n . Finally, as a side result, we obtain a simple identity involving products of four Littlewood-Richardson coefficients.

The Calogero Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero Sutherland pieces but also combinations of the generators of an underlying af fine (sl(2)) over cap algebra.