A note on Cauchy's formula
arXiv:2202.11175 · doi:10.1016/j.aam.2023.102630
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_{μ\subseteq [K\times N]}(-1)^{|μ|}s_μ\{x\}s_{μ'}\{y\}. \end{align*} As an application of the interpretation, we obtain an expansion of in terms of half plane partitions.
13 pages, no figure. Final version