Cayley'S Ω Process

Tech

What is Cayley's Ω process?

In mathematics, Cayley's Ω process, introduced by Arthur Cayley , 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 For binary forms f in x1, y1 and g in x2, y2 the Ω operator is . The r-fold Ω process Ωr(f, g) on two forms f and g in the variables x and y is then 1. * Convert f to a form in x1, y1 and g to a form in x2, y2 2. * Apply the Ω operator r times to the function fg, that is, f times g in these four variables 3. * 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.

invariant theory

Synonyms

Cayley omega processCayley operatorCayley Ω processCayley-Capelli operatorCayley–Capelli operatorCayley's omega processOmega operatorOmega process

Tech Info

Sources: DBpedia
— Date merged: 11/6/2021, 1:32:53 PM
— Date scraped: 5/20/2021, 6:00:48 PM