Invariant Rings of Finite Groups

Creation

Let \(G\) be a finite matrix or permutation group acting on the polynomial ring \(P = K[x_1, \ldots, x_n]\) 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).

Access

The following functions allow simple access to basic properties of invariant rings.

Group(R): RngInvar -> Grp

Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return the group \(G\).

CoefficientRing(R): RngInvar -> Grp

CoefficientField(R): RngInvar -> Rng

Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return the coefficient field \(K\).

PolynomialRing(R): RngInvar -> RngMPol

Given an invariant ring \(R=K[V]^G\) of the group \(G\) of degree \(n\) over the field \(K\), return the polynomial ring \(P = K[x_1, \ldots, x_n]\) in which the invariants of \(R\) lie. \(P\) has the print names "x1", "x2", etc. – the angle bracket notation or the \(.\) operator should be used to assign the variables of \(P\) to actual Magma variables.

f in R: RngMPol, RngInvar -> FldFunUElt, ModMPolElt

Return whether the polynomial \(f\) is in \(R=K[V]^G\). Note that the parent of \(f\) is always the polynomial ring \(P\), never \(R\), so a true result does not mean that the parent of \(f\) is \(R\).