Cauchy identity for symmetric functions (${\prod }_{i,j}(1-x_i{y}_j{)}^{-1}={\sum }_{\lambda }{s}_{\lambda }(x)s_{\lambda }(y)$)

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 ${\prod }_{i,j}(1+x_i{y}_j)={\sum }_{\lambda }{s}_{\lambda }(x)s_{{\lambda }^{\prime }}(y)$.