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$$\operatorname {Tr} \Omega ^{r}(Q_{1}\otimes \cdots \otimes Q_{n})$$where$$\Omega ={\begin{vmatrix}{\frac {\partial }{\partial x_{11}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{n1}}}&\cdots &{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}$$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.$$\operatorname {Tr} \Omega ^{0}(Q_{1}\otimes \cdots \otimes Q_{n})=\prod _{k}Q_{k}$$The first transvectant is the Jacobian determinant of the n functions.$$\operatorname {Tr} \Omega ^{1}(Q_{1}\otimes \cdots \otimes Q_{n})=\det {\begin{bmatrix}\partial _{k}Q_{l}\end{bmatrix}}$$The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.

When $n=2$, the binary transvectants have an explicit formula:$$\operatorname {Tr} \Omega ^{k}(f\otimes g)=\sum _{l=0}^{k}(-1)^{l}{\binom {k}{l}}\partial _{x}^{k-l}\partial _{y}^{l}f\partial _{y}^{k-l}\partial _{l}^{l}g$$which can be more succinctly written as$$f\left({\overleftarrow {\partial _{x}}}\cdot {\overrightarrow {\partial _{y}}}-{\overleftarrow {\partial _{y}}}\cdot {\overrightarrow {\partial _{x}}}\right)^{k}g$$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 ()—All polynomial covariants and invariants of any system of binary forms can be expressed as linear combinations of iterated transvectants.

References

• Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1
• Olver, Peter J.; Sanders, Jan A. (2000), "Transvectants, modular forms, and the Heisenberg algebra", Advances in Applied Mathematics, 25 (3): 252–283, CiteSeerX 10.1.1.46.803, doi:10.1006/aama.2000.0700, ISSN 0196-8858, MR 1783553