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

${\displaystyle \Omega ={\begin{vmatrix}{\frac {\partial }{\partial x_{11}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{n1}}}&\cdots &{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}.}$

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

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.