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

The theorem states that the ring of ${\displaystyle GL(V)}$-invariant polynomial functions on ${\displaystyle {V^{*}}^{p}\times V^{q}}$ is generated by the functions ${\displaystyle \langle \alpha _{i}|v_{j}\rangle }$, where ${\displaystyle \alpha _{i}}$ are in ${\displaystyle V^{*}}$ and ${\displaystyle v_{j}\in V}$.[3]

Second fundamental theorem for general linear group

Let V, W be finite dimensional vector spaces over the complex numbers. Then the only ${\displaystyle \operatorname {GL} (V)\times \operatorname {GL} (W)}$-invariant prime ideals in ${\displaystyle \mathbb {C} [\operatorname {hom} (V,W)]}$ are the determinant ideal ${\displaystyle I_{k}=\mathbb {C} [\operatorname {hom} (V,W)]D_{k}}$ generated by the determinants of all the ${\displaystyle k\times k}$-minors.[4]

Notes

1. Procesi, Ch. 9, § 1.4. harvnb error: no target: CITEREFProcesi (help)
2. Procesi, Ch. 13 develops this theory. harvnb error: no target: CITEREFProcesi (help)
3. Procesi, Ch. 9, § 1.4. harvnb error: no target: CITEREFProcesi (help)
4. Procesi, Ch. 11, § 5.1. harvnb error: no target: CITEREFProcesi (help)

References

• Ch. II, § 4. of E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985. MR0770932
• Artin, Michael (1999). "Noncommutative Rings" (PDF).
• 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
• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255