Cayley's Ω process
# 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 $n^2$ 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 $x_1,$ $y_1$ and g in $x_2,$ $y_2$ 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 $x_1,$ $y_1$ and g to a form in $x_2,$ $y_2$
* Apply the Ω operator r times to the function fg, that is, f times g in these four variables
* Substitute x for $x_1$ and $x_2,$ y for $y_1$ and $y_2$ 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.

