Cayley's Ω process

{{About|the mathematical process|the industrial OMEGA process|OMEGA process}}

In mathematics, Cayley's Ω process, introduced by {{harvs|txt|authorlink=Arthur Cayley|first=Arthur |last=Cayley|year=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

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

For binary forms f in x₁, y₁ and g in x₂, y₂ the Ω operator is $\frac{\partial^2 fg}{\partial x_1 \partial y_2} - \frac{\partial^2 fg}{\partial x_2 \partial y_1}$ . 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 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 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

{{harvtxt|Weyl|1946}} used to find generators for the invariants of various classical groups acting on natural polynomial algebras.

{{harvtxt|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.

References

{{citation|first=Arthur|last=Cayley|title=On linear transformations|journal=Cambridge and Dublin Mathematical Journal|volume=1|year=1846|pages=104–122|url=https://books.google.com/books?id=PBcAAAAAMAAJ&dq=Cambridge+and+Dublin+mathematical+journal+1846&pg=PR3}} Reprinted in {{citation|last=Cayley|title=The collected mathematical papers|volume=1|year=1889|publisher=Cambridge University press|place=Cambridge|pages=95–112}}
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{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The Classical Groups: Their Invariants and Representations | url=https://books.google.com/books?id=zmzKSP2xTtYC | accessdate=26 March 2007 | publisher=Princeton University Press | isbn=978-0-691-05756-9 | mr=0000255 | year=1946}}

Category:Invariant theory