Projective Indecomposable Modules

For a finite group G and a finite field K, the projective indecomposable K[G]-modules are in one-one correspondence with the irreducible K[G]-modules, where the projective indecomposable module P corresponding to the irreducible module I has the property that Socle(P) and P/JacobsonRadical(P) are both isomorphic to I.

Magma functions for the construction of the irreducible modules are described in Subsection Enumerating All Irreducible Modules. The functions described in this section may be used to construct the corresponding projective indecomposables for finite groups of moderate order -- up to around 109, depending on the example. Large-dimensional projective indecomposable modules can sometimes be constructed by using condensation techniques, which allow many of the necessary computations to be carried out in condensed modules, which may have significantly smaller dimension than their standard uncondensed equivalents. As with the computation of irreducible modules, these methods work best for permutation groups and PC-groups.

The verbose flag "KGModule" may be set to 1 or 2 to show details of the computations.

ProjectiveIndecomposableDimensions(G, K) : Grp, FldFin -> SeqEnum
Let G be a finite group and K a finite field. This intrinsic returns the K-dimensions of the projective indecomposable K[G]-modules corresponding to the irreducible K[G]-modules as constructed by the Intrinsic IrreducibleModules(G, K). (These dimensions can be computed quickly from the Brauer characters of the irreducible modules, using the Cartan matrix).
ProjectiveIndecomposableModule(I: parameters) : ModGrp -> ModGrp
Let G be a finite group and let K be a finite field. This intrinsic constructs and returns the projective indecomposable K[G]-module P corresponding to the irreducible K[G]-module I. Note that Socle(P) and P/JacobsonRadical(P) are both isomorphic to I. 
     condensation: BoolElt               Default: false
If set to true, then an attempt is made to find a subgroup of G which allows computations to be carried out using condensed versions of the modules involved.
ProjectiveIndecomposableModules(G, K: parameters) : Grp, FldFin -> SeqEnum
Let G be a finite group and let K be a finite field. This intrinsic constructs and returns the complete list of projective indecomposable K[G]-modules corresponding to the irreducible K[G]-modules as returned by the IrreducibleModules(G,K) intrinsic. 
     condensation: BoolElt               Default: false
If set to true, then an attempt is made to find a subgroup of G which allows computations to be carried out in condensed versions of the modules involved.
PIMBlocks(CM) : AlgMatElt -> SeqEnum
PIMBlocks(I, P) : SeqEnum, SeqEnum -> SeqEnum
Let G be a finite group and let K be a finite field. This intrinsic partitions the projective indecomposable KG-modules (PIMs) into blocks. There are two versions of PIMBlocks, distinguished by argument types. The first version works off the Cartan matrix which if available is very fast. The second version has as arguments sequences I and P containing the irreducible modules and PIMs for G, respectively. This version computes the constituents of each projective indecomposable. Consequentally, as the dimension of the PIMs increases, this can be very time consuming. In each case the blocks are returned as a sequence of sequences of the numbers of those PIMs that lie in given block.
ProjectiveCover(M) : ModGrp -> ModGrp, ModMatGrpElt
Given a K[G]-module M, where K is a finite field, compute the projective cover P of M together with a K[G]-module epimorphism P -> M returned as a matrix.

If Pi is the projective indecomposable K[G]-module corresponding to the irreducible K[G]-module Ii and M/JacobsonRadical(M) is isomorphic to direct-sum j=1t Iij then P is isomorphic to direct-sum j=1t Pij.

InjectiveHull(M) : ModGrp -> ModGrp, ModMatGrpElt
Given a K[G]-module M, where K is a finite field, compute the injective hull I of M together with a K[G]-module monomorphism M -> I returned as a matrix.

If Pi is the projective indecomposable K[G]-module corresponding to the irreducible K[G]-module Ii and Socle(M) is isomorphic to direct-sum j=1t Iij then I is isomorphic to direct-sum j=1t Pij.

Example ModAlg_Projective Indecomposables (H97E29)

We compute the projective indecomposable modules for the alternating group of degree 8 over the field of order 2. 
> G := Alt(8);
> K := GF(2);
> IrreducibleModules(G, K);
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 64 over GF(2)
]
> ProjectiveIndecomposableDimensions(G, K);
[ 448, 192, 192, 320, 320, 192, 192, 64 ]
> time proj := ProjectiveIndecomposables(G, K);
Time: 22.070
> proj;
[
    GModule of dimension 448 over GF(2),
    GModule of dimension 192 over GF(2),
    GModule of dimension 192 over GF(2),
    GModule of dimension 320 over GF(2),
    GModule of dimension 320 over GF(2),
    GModule of dimension 192 over GF(2),
    GModule of dimension 192 over GF(2),
    GModule of dimension 64 over GF(2)
]
> CompositionFactors(proj[1]);
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 20 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 6 over GF(2),
    GModule of dimension 14 over GF(2),
    GModule of dimension 1 over GF(2)
]
V2.28, 13 July 2023