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$$\mathrm{Tr}{\mathrm{\Omega }}^r(Q_1\otimes \cdots \otimes Q_n)$$where$$\mathrm{\Omega }=\left|\begin{array}{ccc}\frac{\partial }{\partial x_{11}}& \cdots & \frac{\partial }{\partial x_{1n}}\\ ⋮& \ddots & ⋮\\ \frac{\partial }{\partial x_{n1}}& \cdots & \frac{\partial }{\partial x_{nn}}\end{array}\right|$$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.$$\mathrm{Tr}{\mathrm{\Omega }}^0(Q_1\otimes \cdots \otimes Q_n)=\prod _k{Q}_k$$The first transvectant is the Jacobian determinant of the n functions.$$\mathrm{Tr}{\mathrm{\Omega }}^1(Q_1\otimes \cdots \otimes Q_n)=\det \left[\begin{array}{c}{\partial }_k{Q}_l\end{array}\right]$$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:$$\mathrm{Tr}{\mathrm{\Omega }}^k(f\otimes g)=\sum _{l=0}^k(-1{)}^l(\genfrac{}{}{0ex}{}{k}{l})\underset{x}{\overset{k-l}{\partial }}\underset{y}{\overset{l}{\partial }}f\underset{y}{\overset{k-l}{\partial }}\underset{l}{\overset{l}{\partial }}g$$which can be more succinctly written as$$f{(\overleftarrow{{\partial }_x}\cdot \overrightarrow{{\partial }_y}-\overleftarrow{{\partial }_y}\cdot \overrightarrow{{\partial }_x})}^{k}g$$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, doi:10.1006/aama.2000.0700, ISSN 0196-8858