On computing the polynomial invariants of a finite group

Abstract

Let p be a real representation of a finite group G as n×n matrices and P(p)G the ring of polynomial invariants associated with p(G) One way to describe P(p)G is as a direct sum . Given that such a good polynomial basis is known for P(p)G . we will show how to construct good polynomial bases for other polynomial rings associated with P(p)G : P(p)H where H is a subgroup of G where σ is another real representation of G, and . We will make sense of the notion of good polynomial basis for relative invariants and show how to construct the same for the representation is the representation gotten from ρ by twisting it by the linear representation . If P(ρ) is the ring of all polynomials associated with ρ(G), then those features of the structure of P(ρ) as a graded G-algebra -needed for the constructions above - will also be developed by extending classical results about the ideal in P(ρ) generated by the invariants, about G-harmonic polynomials and about polarization.