Representations of a reductive algebraic group whose algebras of invariants are complete intersections. (English) Zbl 0575.20036

Nakajima, Hatuhisa

J. Reine Angew. Math. 367, 115-138 (1986).

A linear representation \(\rho\) of a complex reductive algebraic group G over \({\mathbb{C}}\) is said to be COCI, if the algebra of invariant polynomial functions on the space of \(\rho\) under the action of G is a complete intersection. We give a necessary condition for \(\rho\) to be COCI, and using this, show that, if G is simple and simply connected and \(\rho\) is COCI, then \(\rho\) is coregular or one of \((Sym^ 5\Phi,SL_ 2)\), \((Sym^ 6\Phi,SL_ 2)\), \((Sym^ 3\Phi,SL_ 4)\) and \((\Phi \cdot \Lambda^ 2\Phi,SL_ 4)\).

MSC:

20G05	Representation theory for linear algebraic groups
20G20	Linear algebraic groups over the reals, the complexes, the quaternions
15A72	Vector and tensor algebra, theory of invariants

Keywords:

complex reductive algebraic group; invariant polynomial functions; complete intersection