Transvectant Explained

In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process.

Definition

If Q₁,...,Q_n are functions of n variables x = (x₁,...,x_n) and r ≥ 0 is an integer then the r^th transvectant of these functions is a function of n variables given by

tr \Omega ^{r(Q} _{1 ⊗ …} ⊗ Q _{n)} where Ω is Cayley's Ω process, the tensor product means take a product of functions with different variables x¹,..., xⁿ, and tr means set all the vectors x^k equal.

Examples

The zeroth transvectant is the product of the n functions.

The first transvectant is the Jacobian determinant of the n functions.

The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.