Construction of G-modules

This section describes how one can create a finite-dimensional G-module corresponding to the action of a finite group G on a polynomial ring P. There are two ways one can create a finite-dimensional action: the action on the space of homogeneous polynomials of a fixed degree, or the action on the quotient space of polynomials by a zero-dimensional ideal (so the quotient has finite-dimension as a vector space). The functions in this section are also found in the chapter on general modules but are also included here since they are useful in Invariant Theory.

GModule(G, P, d) : Grp, RngMPol, RngIntElt -> ModGrp, Map, @ RngMPolElt @
Given a finite permutation or matrix group G of degree n, a polynomial ring P=K[x1, ..., xn] over a field K, and a non-negative integer d, create the K[G]-module M corresponding to the action of G on the space of homogeneous polynomials of degree d of the polynomial ring P. The function also returns the isomorphism f between the space of homogeneous polynomials of degree d of P and M, together with an indexed set of monomials of degree d of P which correspond to the columns of M.

GModule(G, I, J) : Grp, RngMPol, RngMPol -> ModGrp, Map, @ RngMPolElt @
Given a finite permutation or matrix group G of degree n, an ideal I of a multivariate polynomial ring P=K[x1, ..., xn] over a field K, and a zero-dimensional subideal J of I, create the K[G]-module M corresponding to the action of G on the finite-dimensional quotient I/J. The function also returns the isomorphism f between the quotient space I/J and M, together with an indexed set of monomials of P, forming a (vector space) basis of I/J, and which correspond to the columns of M.

GModule(G, Q) : Grp, RngMPolRes -> ModGrp, Map, @ RngMPolElt @
Given a finite permutation or matrix group G of degree n, and a finite-dimensional quotient ring Q=I/J of a multivariate polynomial ring P=K[x1, ..., xn] over a field K, create the K[G]-module M corresponding to the action of G on the finite-dimensional quotient Q. The function also returns the isomorphism f between the quotient ring Q and M, together with an indexed set of monomials of P, forming a (vector space) basis of Q, and which correspond to the columns of M.

Example RngInvar_GModule (H117E4)

We demonstrate simple uses of the GModule function. 
> q := 5;
> K := GF(q);
> G := GL(3, K);
> P<x, y, z> := PolynomialRing(K, 3);
> I := ideal< P | x^5 - x,y^5 - y, z^5 - z >;
> Q, rho := quo< P | I >;
> f := x^3 + x^2*y + y^3;
> M, phi:= GModule(G, P, I);
> Constituents(M);
[
    GModule of dimension 1 over GF(5),
    GModule of dimension 3 over GF(5),
    GModule of dimension 3 over GF(5),
    GModule of dimension 6 over GF(5),
    GModule of dimension 6 over GF(5),
    GModule of dimension 10 over GF(5),
    GModule of dimension 10 over GF(5),
    GModule of dimension 15 over GF(5),
    GModule of dimension 15 over GF(5),
    GModule of dimension 18 over GF(5),
    GModule of dimension 18 over GF(5),
    GModule of dimension 19 over GF(5)
]
> N := sub<M | phi(f)>;
> N;
GModule N of dimension 10 over GF(5)
> M5 := GModule(G, P, 5);
> M5;
GModule M5 of dimension 21 over GF(5)
> Constituents(M5);
[
    GModule of dimension 3 over GF(5),
    GModule of dimension 18 over GF(5)
]
V2.28, 13 July 2023