A note on Cauchy's formula

Abstract

We use the correlation functions of vertex operators to give a proof of Cauchy's formula$\prod _{i=1}^K\prod _{j=1}^N(1-x_i{y}_j)=\sum _{\mu \subseteq [K\times N]}{(-1)}^{\left|\mu \right|}{s}_{\mu }\left\{x\right\}{s}_{{\mu }^{\prime }}\left\{y\right\}.$ As an application of the interpretation, we obtain an expansion of ${\prod }_{i=1}^{\infty }{(1-q^i)}^{i-1}$ in terms of half plane partitions.