First and second fundamental theorems of invariant theory


In algebra, the first and second fundamental theorems of invariant theory concern the generators and the relations of the ring of invariants in the ring of polynomial functions for classical groups. The theorems are among the most important results of invariant theory.
Classically the theorems are proved over the complex numbers. But characteristic-free invariant theory extends the theorems to a field of arbitrary characteristic.

First fundamental theorem

The theorem states that the ring of -invariant polynomial functions on is generated by the functions, where are in and.

Second fundamental theorem for general linear group

Let V, W be finite dimensional vector spaces over the complex numbers. Then the only -invariant prime ideals in are the determinant ideal
generated by the determinants of all the -minors.