# 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 ${\bf F}_{3}$.

```magma
> 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

```