Cayley's Ω process explained

In mathematics, Cayley's Ω process, introduced by, 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} (\partial)/(\partial x _{11}) & … & (\partial)/(\partial x _{1n}) \ \vdots& \ddots & \vdots\ (\partial)/(\partial x _{n1}) & … & (\partial)/(\partial x _{nn}) \end{vmatrix}.

For binary forms f in x₁, y₁ and g in x₂, y₂ the Ω operator is

(\partial ^{2} fg)/(\partial x _{1} \partial y _{2}) - (\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.

Applications

Cayley's Ω process appears in Capelli's identity, which used to find generators for the invariants of various classical groups acting on natural polynomial algebras.

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

• Reprinted in