Mathematical process
In mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846), is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action.
As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by the determinant
![{\displaystyle \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}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
- Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen, 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831, S2CID 179177713
- Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American Mathematical Society, 313 (2), American Mathematical Society: 539–570, doi:10.1090/S0002-9947-1989-0986027-X, ISSN 0002-9947, JSTOR 2001418, MR 0986027
- Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1
- Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, ISBN 978-3-211-82445-0, MR 1255980
- Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255, retrieved 26 March 2007