Cayley's Ω 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 x_ij, the omega operator is given by the determinant

\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}}.

For binary forms f in x₁, y₁ and g in x₂, y₂ the Ω operator is ${\frac {\partial ^{2}fg}{\partial x_{1}\partial y_{2}}}-{\frac {\partial ^{2}fg}{\partial x_{2}\partial y_{1}}}$ . The r-fold Ω process Ω^r(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 Ω^r(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