ABSTRACT
The roots of invariant theory can be traced back at least to Gauss who, in his Disquisitiones Arithmeticae, showed that the discriminant of a binary form in two indeterminates upon a linear change of variables changes with the square of the determinant of the change of variables. This chapter describes algorithms that arise in the invariant theory of algebraic groups and mostly concerned with concerned with reductive groups. An algebraic group G is called linearly reductive if every finite-dimensional rational representation of G is completely reducible. The chapter considers the Harm Derksen ideal, which plays a main role in Derksen’s algorithm for computing generators of the invariant ring of a reductive algebraic group. It also appears in an algorithm for computing generators of the invariant field corresponding to a rational action of an algebraic group. The input to the algorithm consists of a basis B of the fixed Cartan subalgebre nsi onss can be a basis of a vector space.
