Classical symmetric functions in superspace

Abstract

We present the basic elements of a generalization of symmetric function theory 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 on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.

21 pages, this supersedes the first part of math.CO/0412306

References in corpus (4)

• Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
• Jack polynomials in superspace
• Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model
• Jack superpolynomials, superpartition ordering and determinantal formulas

Cited by in corpus (12)

• Orthogonality of Jack polynomials in superspace
• Superconformal field theory and Jack superpolynomials
• Evaluation and normalization of Jack superpolynomials
• Macdonald polynomials in superspace: conjectural definition and positivity conjectures
• The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
• Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
• Schur Superpolynomials: Combinatorial Definition and Pieri Rule
• Super-Whittaker vector at c=3/2
• Bernstein operators and super-Schur functions: combinatorial aspects
• N>=2 symmetric superpolynomials
• Symmetric functions in noncommuting variables in superspace
• On the Hopf superalgebra of symmetric functions in superspace