First and second fundamental theorems of invariant theory

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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[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 determinant of all the -minors.[4]


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


  • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
  • Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.
  • Hanspeter Kraft and Claudio Procesi, Classical Invariant Theory, a Primer