Cayley's Ω process

 ( Invariant Theory ) 

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} \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 $\backslash frac\{\backslash partial^2\; fg\}\{\backslash partial\; x\_1\; \backslash partial\; y\_2\}\; -\; \backslash frac\{\backslash partial^2\; fg\}\{\backslash partial\; x\_2\; \backslash partial\; y\_1\}$. 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.

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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 .

• Reprinted in

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