A note on Cauchy's formula

TL;DR

This paper provides a proof of Cauchy's formula using vertex operator correlation functions and applies it to expand a product involving partitions, linking algebraic identities with combinatorial structures.

Contribution

It introduces a novel proof of Cauchy's formula via vertex operator correlation functions and derives an expansion related to half plane partitions.

Findings

Proof of Cauchy's formula using vertex operators

Expansion of an infinite product in terms of half plane partitions

Connection between algebraic identities and combinatorial structures

Abstract

We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{\mu\subseteq [K\times N]}(-1)^{|\mu|}s_{\mu}\{x\}s_{\mu'}\{y\}. \end{align*} As an application of the interpretation, we obtain an expansion of $\prod^\infty_{i=1}(1-q^i)^{i-1}$ in terms of half plane partitions.