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The Construction of 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 were described in Subsection Generic Functions for Finding 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 106, 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
The K-dimensions of the projective indecomposable K[G]-modules corresponding to the irreducible K[G]-modules returned by IrreducibleModules(G,K). (These can be computed quickly from the Brauer characters of the irreducible modules, using the Cartan matrix discussed below.)
ProjectiveIndecomposableModule(I: parameters) : ModGrp -> ModGrp
Construct and return 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 in condensed versions of the modules involved.
ProjectiveIndecomposableModules(G, K: parameters) : Grp, FldFin -> SeqEnum
Construct the complete list of projective indecomposable K[G]-modules corresponding to the irreducible K[G]-modules returned by IrreducibleModules(G,K).

     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.

Example ModGrp_Projective Indecomposables (H90E19)

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)
]

CartanMatrix(G, K) : Grp, FldFin -> AlgMatElt
Let k be the number of irreducible K[G]-modules. The Cartan Matrix C for G over K is a k x k matrix of integers, in which the entry Cij is equal to the number of times that the j-th irreducible K[G]-module is a constituent of the i-th projective indecomposable K[G]-module. This can be computed quickly from the Brauer characters of the irreducible K[G]-modules, and is used in ProjectiveIndecomposableDimensions.

(Note that, unlike the absolute Cartan matrix discussed below, C need not be symmetric.)

AbsoluteCartanMatrix(G, K) : Grp, FldFin -> AlgMatElt
This is the Cartan matrix of G over an extension L of K that is large enough for all irreducible L[G]-modules to be absolutely irreducible. Its rows and columns correspond to the K[G]-modules returned by AbsolutelyIrreducibleModules(G,K).

It is a symmetric matrix with integer entries, and is equal to the Cartan matrix for G in the characteristic p of K, as defined in textbooks on modular representation theory.

DecompositionMatrix(G, K) : Grp, FldFin -> AlgMatElt
The decomposition matrix D of G in the characteristic p of K, as defined in textbooks on modular representation theory. The entry Dij is equal to the number of times that the j-th absolutely irreducible K[G]-module occurs as a constituent of the i-th ordinary irreducible G-module over the complex numbers reduced modulo p. Note that D T D is equal to the absolute Cartan matrix.
ProjectiveCover(M) : ModGrp -> ModGrp, ModMatGrpElt
Compute the projective cover P of K[G]-module 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.

CohomologicalDimension(M, n) : ModGrp, n -> RngIntElt
For K[G]-module M (with K a finite field and G a finite group), compute and return the K-dimension of the cohomology group Hn(G, M) for n ≥0. For n=0 and 1, this is carried out by using the functions described in Chapter COHOMOLOGY AND EXTENSIONS. For n ≥2, it is done recursively using projective covers and dimension shifting to reduce to the case n=1. (In particular, for n=2, the method is different from that employed by the corresponding function for a cohomology module described in Chapter COHOMOLOGY AND EXTENSIONS.)
CohomologicalDimensions(M, n) : ModGrp, n -> RngIntElt
For K[G]-module M (with K a finite field and G a finite group), compute and return the sequence of K-dimensions of the cohomology groups Hk(G, M) for 1 ≤k ≤n. On account of the recursive method used, this is quicker than computing them individually.

Example ModGrp_Cohomological Dimension (H90E20)

We compute the cohomology of the irreducible modules for Alt(8) over the field of order 2 computed in the previous example.

> G := Alt(8);
> K := GF(2);
> irr :=  IrreducibleModules(G, K);            
> [ CohomologicalDimension(I, 1) : I in irr ];
[ 0, 0, 0, 1, 1, 1, 1, 0 ]
> time [ CohomologicalDimension(I, 2) : I in irr ];
[ 1, 1, 1, 0, 2, 0, 0, 0 ]
Time: 0.440
> time [ CohomologicalDimension(I, 3) : I in irr ];
[ 2, 1, 1, 1, 1, 1, 1, 0 ]
Time: 15.070
> time [ CohomologicalDimension(I, 4) : I in irr ];
[ 2, 1, 1, 2, 3, 2, 2, 0 ]
Time: 99.500
> time CohomologicalDimensions(irr[1], 6);
[ 0, 1, 2, 2, 3, 6 ]
Time: 139.270
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