In algebra, the first and second fundamental theorems of invariant theory concern the generators and 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). 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 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} .
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.
Notes
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 {{citation}}: ISBN / Date incompatibility (help)