About Gordan's algorithm for binary forms

TL;DR

This paper revisits Gordan's algorithm for binary forms, using a modern, graphical approach based on equivariant homomorphisms to compute covariant bases, including new minimal bases for specific representations.

Contribution

It introduces a new graphical method for Gordan's algorithm, enabling the computation of minimal covariant bases for complex binary form representations.

Findings

First minimal covariant basis for S_6 ⊕ S_4

First minimal covariant basis for S_6 ⊕ S_4 ⊕ S_2

Enhanced understanding of Gordan's algorithm through graphical methods

Abstract

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 $\Sn{6}\oplus\Sn{4}\oplus \Sn{2}$.