Bernstein operators and super-Schur functions: combinatorial aspects

Abstract

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.