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 (roughly the first concerns the generators and the second the relations).[1] 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.[2]

First fundamental theorem for

edit

The theorem states that the ring of  -invariant polynomial functions on   is generated by the functions  , where   are in   and  .[3]

Second fundamental theorem for general linear group

edit

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.[4]

Notes

edit
  1. ^ Procesi 2007, Ch. 9, § 1.4.
  2. ^ Procesi 2007, Ch. 13 develops this theory.
  3. ^ Procesi 2007, Ch. 9, § 1.4.
  4. ^ Procesi 2007, Ch. 11, § 5.1.

References

edit
  • Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 978-0-387-26040-2. OCLC 191464530.

Further reading

edit