This page will investigate RSK for classical groups and Howe duality at the crystal level. \[ \begin{array} {|c|c|} \hline \phi_i (x \otimes y \otimes z) & \begin{array} {c} f_i (x \otimes y \otimes z)\\ \text{first applicable value} \end{array} \\ \hline \phi_i (z) & x \otimes y \otimes f_i (z)\\ \hline \phi_i (y) + \phi_i (z) - \varepsilon_i (z) & x \otimes f_i (y) \otimes z\\ \hline \phi_i (x) + \phi_i (y) + \phi_i (z) - \varepsilon_i (y) - \varepsilon_i (z) & f_i (x) \otimes y \otimes z\\ \hline \end{array} \] Thus $\phi_i (x \otimes y \otimes z)$ is the maximum of the values in the first column. If it appears more than once, $f_i (x \otimes y \otimes z)$ to be second column value from the first time the maximum is attained. \begin{eqnarray*} \phi_i (x \otimes y) & = & \max (\phi_i (y), \phi_i (x) + \phi_i (y) - \varepsilon_i (y)),\\ \varepsilon_i (x \otimes y) & = & \max (\varepsilon_i (x), \varepsilon_i (x) + \varepsilon_i (y) - \phi_i (x)) \end{eqnarray*}