JOURNAL OF
ALGEBRAIC
COMBINATORICS

Home > Vol 24, No 2 (2006)

Patrick Desrosiers¹ , Luc Lapointe² and Pierre Mathieu³

¹The University of Melbourne Department of Mathematics and Statistics Parkville Australia 3010 Parkville Australia 3010
²Universidad de Talca Instituto de Matemática y Física Casilla 747 Talca Chile Casilla 747 Talca Chile
³Université Laval Département de physique, de eénie physique et d'optique Québec Canada G1K 7P4 Québec Canada G1K 7P4

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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