OMEGA process

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.

As a partial differential operator acting on functions of n² variables x_ij, the omega operator is given by the determinant

For binary forms f in x₁, y₁ and g in x₂, y₂ the Ω operator is ${\displaystyle {\frac {\partial ^{2}fg}{\partial x_{1}\partial y_{2}}}-{\frac {\partial ^{2}fg}{\partial x_{2}\partial y_{1}}}}$. The r-fold Ω process Ω^r(f, g) on two forms f and g in the variables x and y is then

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