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arXiv · 2202.11175

A note on Cauchy's formula

Naihuan Jing·Zhijun Li
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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 $\prod^\infty_{i=1}(1-q^i)^{i-1}$ in terms of half plane partitions.

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Naihuan Jing, Zhijun Li. 2023-10-12. A note on Cauchy's formula. https://doi.org/10.1016/j.aam.2023.102630

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