Let G be a finite matrix or permutation group acting on the polynomial ring P = K[x_1, ..., x_n] over the field K. Magma allows the construction of the invariant ring R=K[V]^G. The invariant ring R is a special structure which contains references to the group G and polynomial ring P. When the invariant ring R is created using the InvariantRing function, no explicit calculations are done until specifically invoked (e.g. by the PrimaryInvariants function). The elements of R are the polynomials of P which are invariant under the action of G. Note that the parent of such polynomials is still P -- the invariant ring R is just a special structure which contains all the information about the invariant ring. The category of invariant rings is RngInvar.

InvariantRing(G) : GrpMat -> RngInvar

InvariantRing(G, K) : GrpPerm, Fld -> RngInvar

Construct the invariant ring R=K[V]^G of the finite matrix or permutation group G over the field K. For a matrix group G, G alone should be supplied, while for a permutation group G, G should be supplied, together with the field K. The appropriate multivariate polynomial ring P is automatically constructed. No other explicit calculations are done (e.g. computation of primary invariants).