Transvectant
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In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process.
Definition
If Q1,...,Qn are functions of n variables x = (x1,...,xn) and r ≥ 0 is an integer then the rth transvectant of these functions is a function of n variables given by
- [math]\displaystyle{ tr \Omega^r(Q_1\otimes\cdots \otimes Q_n) }[/math]
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 xk 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.
Footnotes
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
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