In mathematics, Cayley's Ω process, introduced by , is a relatively invariant differential operator on the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

, that is used to construct invariants of a

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

. As a

partial differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

acting on functions of ''n''² variables ''x''_''ij'', the omega operator is given by the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

\Omega = \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac  \end.

For binary forms ''f'' in ''x''₁, ''y''₁ and ''g'' in ''x''₂, ''y''₂ the Ω operator is

\frac - \frac

. 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 ''x''₁, ''y''₁ and ''g'' to a form in ''x''₂, ''y''₂ # Apply the Ω operator ''r'' times to the function ''fg'', that is, ''f'' times ''g'' in these four variables # Substitute ''x'' for ''x''₁ and ''x''₂, ''y'' for ''y''₁ and ''y''₂ 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 In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n ...

, 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 In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountere ...

of the special linear group. Cayley's Ω process is used to define transvectants.

References

* Reprinted in * * * * * {{DEFAULTSORT:Cayley's Omega process Invariant theory