Symmetric functions, codes of partitions and the KP hierarchy

TL;DR

This paper introduces a new explicit formula for Bernstein operators acting on Schur functions using partition codes, and provides a simplified proof of a classical KP hierarchy result via symmetric Plucker relations.

Contribution

It presents a novel explicit formula for Bernstein operators in terms of partition codes and offers a more symmetric, simplified proof of a key KP hierarchy theorem.

Findings

Explicit Bernstein operator formula in partition code notation

Simplified proof of KP hierarchy classical result

Symmetrical restatement of Plucker relations

Abstract

We consider an operator of Bernstein for symmetric functions, and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plucker relations for Schur function coefficients in a tau function for the hierarchy. This proof is especially compact because of a restatement that we give for the Plucker relations that is symmetrical in terms of partition code notation.