Classical symmetric functions in superspace

Details Classical symmetric functions in superspace Classical symmetric functions in superspace

2005-09-19

arxiv.org/abs/math/0509408v1

Mathematics

Combinatorics

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

Scientific paper

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.

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