Cayley's Ω process [SEP] named after
Cayley's Ω process
In mathematics, Cayley's Ω process, introduced by , 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 "n" variables "x", the omega operator is given by the determinant
For binary forms "f" in "x", "y" and "g" in "x", "y" the Ω operator is formula_2. The "
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 formula_1. It can be used to relate an invariant "ƒ" to the invariant Ω"ƒ", where Ω is Cayley's Ω process.
Statement.
Suppose that "x" for "i","j" = 1...,"n" are commuting variables. Write