= Cauchy identity for symmetric functions
{c}
{title2=$\prod_{i,j}(1-x_iy_j)^{-1}=\sum_\lambda s_\lambda(x)s_\lambda(y)$}
= Cauchy identity
{c}
{synonym}
Expand each factor as a <geometric series> indexed by a nonnegative <matrix>. The <RSK correspondence> preserves its row and column contents, changing the sum to pairs of <Semistandard Young tableaux> of the same shape and proving the displayed formal identity. Its kernel is also $\sum_\lambda h_\lambda(x)m_\lambda(y)=\sum_\lambda p_\lambda(x)p_\lambda(y)/z_\lambda$. These are dual-basis kernels for the <Hall inner product of symmetric functions>, so the Schur expansion proves orthonormality. Applying the <symmetric-function involution> in one alphabet gives the dual identity $\prod_{i,j}(1+x_iy_j)=\sum_\lambda s_\lambda(x)s_{\lambda'}(y)$.
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