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 bywhereis 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 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.
When , the binary transvectants have an explicit formula:[1]which can be more succinctly written aswhere 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.
References
- ^ Olver 1999, p. 88.
- ^ Olver 1999, p. 90.
- 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
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