The Module of an Invariant Ring

Let \(R=K[V]^G\) be the invariant ring of a finite group \(G\) over the field \(K\) and suppose the degree of \(G\) is \(n\). Suppose also that primary invariants \(\{ f_1, \ldots, f_n \}\) for \(R\) have been constructed, together with minimal secondary invariants \(S = \{ g_1, \ldots, g_m \}\) for \(R\) with respect to these primary invariants. (These secondary invariants may possess non-trivial module syzygies.) Then \(R\) can be considered as a module over the algebra \(A = K[f_1, \ldots, f_n]\) with the minimal (module) generating set \(S\). To compute with this module structure of \(R\) easily, Magma automatically constructs the graded multivariate polynomial algebra \(A' = K[t_1, \ldots, t_n]\) (with the weighted degree of the variable \(t_i\) defined to be the degree of \(f_i\)) which is isomorphic to \(A\), and then constructs the graded module \(M = A'^m/Q\) over \(A'\) with the quotient relations \(Q\) given by the syzygies of the \(g_i\) (and with the weighted degree of column \(i\) equal to the degree of \(g_i\)). The algebra \(A'\) is isomorphic to \(A\) under the map \(t_i \mapsto f_i\), and the module \(M\) is isomorphic to \(R\) (considered as a module) under the map \(M.i \mapsto g_i\) (extended by the isomorphism from \(A'\) onto \(A\)). (See the chapter on modules over \(K[x_1, \ldots, x_n]\) for details on how to compute with the module \(M\) and an explanation of quotient relations, the unit vectors \(M.i\), etc.) Once the module \(M\) is created, together with the isomorphism \(f: R \rightarrow M\), one can apply \(f\) to a general element \(h\) of \(R\) to obtain the element of \(M\) corresponding to \(h\). This effectively yields a representation of \(h\) as a sum \(\sum_{i=1}{k} a_i g_i\) with \(a_i \in A\) in terms of the primary and secondary invariants. This representation is also unique up to the relations given by the syzygies of the \(g_i\). When creating the module \(M\), the coefficient ring \(A'\) of \(M\) is assigned the print names "t1", "t2", etc. – the angle bracket notation or the \(.\) operator should be used to assign the variables of \(A'\) to actual Magma variables.

Module(R): RngInvar -> ModMPol, Map

The module \(M\) isomorphic to \(R=K[V]^G\), together with the isomorphism \(f: R \rightarrow M\).

Example: Module (ex-822234)

We create the module \(M\) corresponding to the invariant ring \(R\) of the group \(G\) generated by the 4 by 4 Jordan block over \({\bf F}_{3}\).

> K := GF(3);
> G := MatrixGroup<4,K | [1,0,0,0, 1,1,0,0, 0,1,1,0, 0,0,1,1]>;
> R := InvariantRing(G);
> P<x1,x2,x3,x4> := PolynomialRing(R);
> p := PrimaryInvariants(R);
> s := SecondaryInvariants(R);
> [TotalDegree(f): f in p];
[ 1, 2, 3, 9 ]
> [TotalDegree(f): f in s];
[ 0, 3, 4, 5, 6, 7, 8, 9 ]
> M, f := Module(R);
> M;
Full Quotient Module of degree 8
TOP Order
Column weights: 0 3 4 5 6 7 8 9
Coefficient ring:
    Graded Polynomial ring of rank 4 over GF(3)
    Lexicographical Order
    Variables: t1, t2, t3, t4
    Variable weights: 1 2 3 9
Quotient Relations:
[
    t1[7] + 2*t2[6] + t3[5],
    t1[4] + 2*t2[3] + t3[2]
]
> h := x1^5*x2 + 2*x1^3*x3^3 + 2*x2^6;
> h;
x1^5*x2 + 2*x1^3*x3^3 + 2*x2^6
> m := f(h);
> m;
t1^4*t2[1] + t1^3[2] + t2^3[1]
> // Evaluate in the primaries and secondaries:
> p[1]^4*p[2]*s[1] + p[1]^3*s[2] + p[2]^3*s[1];
x1^5*x2 + 2*x1^3*x3^3 + 2*x2^6

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