Английская Википедия:Cayley's Ω process
Шаблон:Short description Шаблон:About In mathematics, Cayley's Ω process, introduced by Шаблон:Harvs, 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 xij, the omega operator is given by the determinant
- <math>
\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>\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 Ωr(f, g) on two forms f and g in the variables x and y is then
- Convert f to a form in x1, y1 and g to a form in x2, y2
- Apply the Ω operator r times to the function fg, that is, f times g in these four variables
- Substitute x for x1 and x2, y for y1 and y2 in the result
The result of the r-fold Ω process Ωr(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 Шаблон:Harvtxt used to find generators for the invariants of various classical groups acting on natural polynomial algebras.
Шаблон:Harvtxt 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
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