Decomposable Symmetric Tensors

In the space of symmetric tensors $S^d V$, where $V$ is a finite--dimensional vector space on $\mathbb{C}$, the decomposable tensors $f_1 f_2 \cdots f_d$ form an irreducible algebraic subvariety $D$. If $V$ is the dual of a vector space $W$, then $S^d V$ identifies with the forms of degree $d$ on $W$, and the decomposable tensors are just the products of linear forms. An interesting problem is to find equations defing $D$ in $S^d V$. I will give an overview of the known solutions to this problem. Some of these solutions are obtained through classical invariant theory. One such solution is due to Brill and Gordan, another one is due to Gaeta. Another approach consists in working in coordinates. Then one meets objects that are interesting by themselves, the \emph{diagonal invariants of the symmetric group}, that have very nice combinatorial properties that I will present.