Cayley's Ω process

Short description: 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 x_ij, the omega operator is given by the determinant

[math]\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}. }[/math]

For binary forms f in x₁, y₁ and g in x₂, y₂ the Ω operator is [math]\displaystyle{ \frac{\partial^2 fg}{\partial x_1 \partial y_2} - \frac{\partial^2 fg}{\partial x_2 \partial y_1} }[/math]. The r-fold Ω process Ω^r(f, g) on two forms f and g in the variables x and y is then

1. Convert f to a form in x₁, y₁ and g to a form in x₂, y₂
2. Apply the Ω operator r times to the function fg, that is, f times g in these four variables
3. 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, https://books.google.com/books?id=PBcAAAAAMAAJ&dq=Cambridge+and+Dublin+mathematical+journal+1846&pg=PR3  Reprinted in Cayley (1889), The collected mathematical papers, 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 
• Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American Mathematical Society (American Mathematical Society) 313 (2): 539–570, doi:10.1090/S0002-9947-1989-0986027-X, ISSN 0002-9947 
• 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 
• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, https://books.google.com/books?id=zmzKSP2xTtYC, retrieved 26 March 2007