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

Apply the symmetric-function involution in one alphabet to the ordinary Cauchy identity. It changes complete generators to elementary ones and, by the dual Jacobi–Trudi identity, sends $s_{\lambda }$ to $s_{{\lambda }^{\prime }}$. This proves the identity as a formal series and relates pairings with transposed Young shapes to squarefree matrix weights.