Traces, symmetric functions, and a raising operator

TL;DR

This paper introduces a new raising operator extending the Heisenberg algebra, providing a graphical proof of symmetric function relations with applications in invariants of Lax equations and field theories.

Contribution

It develops a novel raising operator framework that generalizes the Heisenberg algebra and offers a simple graphical proof of symmetric function identities.

Findings

Extended Heisenberg algebra with a new operator

Graphical proof of elementary symmetric function relations

Applications in invariants for Lax equations and field theories

Abstract

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator extends the Heisenberg algebra so that the number operator becomes a Lie product. This study is motivated by natural appearance of these polynomials in the theory of invariants for Lax equations and in classical and topological field theories.