Cayley's Ω process
In mathematics, Cayley's Ω process, introduced by
As a
For binary forms f in x1, y1 and g in x2, y2 the Ω operator is . The r-fold Ω process Ωr(f, g) on two forms f and g in the variables x and y is then
- Convert f to a form in x1, y1 and g to a form in x2, y2
- Apply the Ω operator r times to the function fg, that is, f times g in these four variables
- Substitute x for x1 and x2, y for y1 and y2 in the result
The result of the r-fold Ω process Ωr(f, g) on the two forms f and g is also called the r-th transvectant and is commonly written (f, g)r.
Applications
Cayley's Ω process appears in Capelli's identity, which Weyl (1946) used to find generators for the invariants of various classical groups acting on natural polynomial algebras.
Hilbert (1890) used Cayley's Ω process in his proof of finite generation of rings of invariants of the general linear group. His use of the Ω process gives an explicit formula for the Reynolds operator of the special linear group.
Cayley's Ω process is used to define transvectants.
References
- Cayley, Arthur (1846), "On linear transformations", Cambridge and Dublin Mathematical Journal, 1: 104–122 Reprinted in Cayley (1889), The collected mathematical papers, vol. 1, Cambridge: Cambridge University press, pp. 95–112
- S2CID 179177713
- MR 0986027
- ISBN 978-0-521-55821-1
- MR 1255980
- MR 0000255, retrieved 26 March 2007