Transvectant

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$${\displaystyle \operatorname {Tr} \Omega ^{r}(Q_{1}\otimes \cdots \otimes Q_{n})}$$where$${\displaystyle \Omega ={\begin{vmatrix}{\frac {\partial }{\partial x_{11}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{n1}}}&\cdots &{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}}$$is Cayley's Ω process, and the tensor product means take a product of functions with different variables x¹,..., xⁿ, and the trace operator Tr means setting all the vectors x^k equal.

Examples

The zeroth transvectant is the product of the n functions.$${\displaystyle \operatorname {Tr} \Omega ^{0}(Q_{1}\otimes \cdots \otimes Q_{n})=\prod _{k}Q_{k}}$$The first transvectant is the Jacobian determinant of the n functions.$${\displaystyle \operatorname {Tr} \Omega ^{1}(Q_{1}\otimes \cdots \otimes Q_{n})=\det {\begin{bmatrix}\partial _{k}Q_{l}\end{bmatrix}}}$$The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.

When ${\displaystyle n=2}$, the binary transvectants have an explicit formula:^[1]$${\displaystyle \operatorname {Tr} \Omega ^{k}(f\otimes g)=\sum _{l=0}^{k}(-1)^{l}{\binom {k}{l}}\partial _{x}^{k-l}\partial _{y}^{l}f\partial _{y}^{k-l}\partial _{l}^{l}g}$$which can be more succinctly written as$${\displaystyle f\left({\overleftarrow {\partial _{x}}}\cdot {\overrightarrow {\partial _{y}}}-{\overleftarrow {\partial _{y}}}\cdot {\overrightarrow {\partial _{x}}}\right)^{k}g}$$where the arrows denote the function to be taken the derivative of. This notation is used in Moyal product.

Applications

First Fundamental Theorem of Invariant Theory (^[2])—All polynomial covariants and invariants of any system of binary forms can be expressed as linear combinations of iterated transvectants.