Full reducibility and invariants for \(\text{SL}_2(\mathbb{C})\) (Q1594927)

A representation \(V\) of a group \(G\) is called fully reducible if it is a direct sum of irreducible representations. The paper is focused on the case when \(G=\text{SL}_2(\mathbb{C})\) (complex \(2\times 2\) matrices with determinant 1) and \(V\) is any finite-dimensional complex holomorphic representation. The full reducibility in this case is a fact with an interesting history. Details of different proofs are given. More general situations and connections with invariant theory and other problems are also discussed.