Bernstein operators and super-Schur functions: combinatorial aspects

TL;DR

This paper extends Bernstein operators to superspace, defining super vertex operators for super-Schur functions, and provides combinatorial proofs of their recursive construction, with implications for super-KP hierarchy realization.

Contribution

It introduces four families of super Bernstein vertex operators and proves their role in constructing super-Schur functions recursively, advancing combinatorial understanding in superspace.

Findings

Defined four super vertex operator families

Proved recursive construction of super-Schur functions

Suggested realization of super-KP hierarchy in superspace

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.