In mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846), is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action.
The r-fold Ω process Ωr(f, g) on two forms f and g in the variables x and y is then 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. Cayley's Ω process appears in Capelli's identity, which Weyl (1946) used to find generators for the invariants of various classical groups acting on natural polynomial algebras.
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.