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 GL ⁡ ( V ) {\displaystyle \operatorname {GL} (V)}

The theorem states that the ring of GL ⁡ ( V ) {\displaystyle \operatorname {GL} (V)} -invariant polynomial functions on V ∗ p ⊕ V q {\displaystyle {V^{*}}^{p}\oplus V^{q}} is generated by the functions ⟨ α i | v j ⟩ {\displaystyle \langle \alpha _{i}|v_{j}\rangle } , where α i {\displaystyle \alpha _{i}} are in V ∗ {\displaystyle V^{*}} and v j ∈ V {\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 GL ⁡ ( V ) × GL ⁡ ( W ) {\displaystyle \operatorname {GL} (V)\times \operatorname {GL} (W)} -invariant prime ideals in C [ hom ⁡ ( V , W ) ] {\displaystyle \mathbb {C} [\operatorname {hom} (V,W)]} are the determinant ideal I k = C [ hom ⁡ ( V , W ) ] D k {\displaystyle I_{k}=\mathbb {C} [\operatorname {hom} (V,W)]D_{k}} generated by the determinants of all the k × k {\displaystyle k\times k} -minors.[4]

Notes

  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

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

Further reading

  • 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. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
  • 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


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