About Gordan's algorithm for binary forms

In this paper, we present a modern version of Gordan's algorithm on binary forms. Symbolic method is reinterpreted in terms of $\mathsf{SL}_2(\mathbb{C})$--equivariant homomorphisms defined upon Cayley operator and polarization operator. A graphical approach is thus developed to obtain Gordan's ideal, a central key to get covariant bases of binary forms. To illustrate the power of this method, we obtain for the first time a minimal covariant basis for $\Sn{6}\oplus\Sn{4}$ and $\Sn{6}\oplus\Sn{4}\oplus \Sn{2}$.