Symmetric functions in superspace

P. Desrosiers, L. Lapointe, P. Mathieu

Published 2004-12-15Version 1

We construct a generalization of the theory of symmetric functions involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group acting on the sets of commuting and anticommuting variables. We first obtain superspace analogues of a number of standard objects and concepts in the theory of symmetric functions: partitions, monomials, elementary symmetric functions, completely symmetric functions, power sums, involutions, generating functions, Cauchy formulas, and scalar products. We then consider a one-parameter extension of the combinatorial scalar product. It provides the natural setting for the definition of a family of ``combinatorial'' orthogonal Jack polynomials in superspace. We show that this family coincides with that of ``physical'' Jack polynomials in superspace that were previously introduced by the authors as orthogonal eigenfunctions of a supersymmetric quantum mechanical many-body problem. The equivalence of the two families is established by showing that the ``physical'' Jack polynomials are also orthogonal with respect to the combinatorial scalar product. This equivalence is also directly demonstrated for particular values of the free parameter.