MAGMA Computational Algebra System

The indecomposable projective modules are defined from the structure of the algebra and have associated path trees that solve the homomorphism lifting problem.

ProjectiveModule(B, i) : AlgBas, RngIntElt -> ModRng

The i^h projective module of the basic algebra B.

PathTree(B, i) : AlgBas, RngIntElt -> ModRng

The path tree of the i^h projective module of the basic algebra B.

ActionGenerator(B, i) : AlgBas, RngIntElt -> SeqEnum

The sequence of matrices for the generators of the basic algebra B acting on the i^h projective module of B.

IdempotentActionGenerators(B, i) : AlgBas, RngIntElt -> SeqEnum

The sequence of matrices for the idempotent generators of the basic algebra B acting on the i^h projective module of B.

NonIdempotentActionGenerators(B, i) : AlgBas, RngIntElt -> SeqEnum

The sequence of matrices for the nonidempotent generators of the basic algebra B acting on the i^h projective module of B.

Injection(B, i, v) : AlgBas, RngIntElt, ModRngElt -> AlgBasElt

Given a vector v in in the i^h projective module of the basic algebra B, the function returns the image of inclusion of v into B.

Creation

AModule(B, Q) : AlgBas, SeqEnum[AlgMatElt] -> ModRng

Given a basic algebra B and a sequence Q of elements in a matrix algebra the function returns the B-module M on which the generators of B act by multiplication by the corresponding elements of Q.

ProjectiveModule(B, S) : AlgBas, SeqEnum[RngIntElt] -> ModAlg, SeqEnum, SeqEnum

Given a sequence S = [s₁, s₂, ... ], the function returns a projective module which is the direct sum of s₁ copies of the first projective of the algebra B, s₂ copies of the second, etc. It also returns the sequence of inclusions and projections from and to the indecomposable projective modules.

IrreducibleModule(B, i) : AlgBas, RngIntElt -> ModAlg

SimpleModule(B, i) : AlgBas, RngIntElt -> ModAlg

The i^h irreducible module of the algebra B. The module is the quotient of the i^h projective module by its radical.

ZeroModule(B) : AlgBas -> ModAlg

The zero B-module.

RightRegularModule(B) : AlgBas -> ModAlg

The algebra B as a right module over itself. The module is the direct sum of the projectives modules of B.

RegularRepresentation(v) : AlgBasElt -> AlgMatElt

If v is an element of a basic algebra given as a vector in the underlying space, then the function computes the matrix of the action by right multiplication of the element on the algebra.

JacobsonRadical(M) : ModAlg -> ModAlg

The Jacobson radical of the module M.

Socle(M) : ModAlg -> ModAlg

The socle of the module M. The sum of the simple submodules of M.

Access Functions

Algebra(M) : ModAlg -> AlgBas

Given a module M over a basic algebra B, the function returns B.

Dimension(M) : ModAlg -> RngIntElt

The dimension of the module M over the its base ring.

Action(M) : ModAlg -> AlgMat

The matrix algebra of the action of the algebra of M on M.

Predicates

The following functions return a boolean value.

IsSemisimple(M) : ModAlg -> BoolElt, SeqEnum

Returns true if the module M is a semisimple module and false otherwise. If true, then the function also returns a list of the ranks of the primitive idempotents of the algebra. This is also a list of the multiplicities of the simple modules of the algebra as composition factors in a composition series for the module.

IsProjective(M) : ModAlg -> BoolElt, SeqEnum

Returns true if the module M is projective. The function also returns a sequence of multiplicities of the standard projective modules as direct summands of the projective cover of M.

IsInjective(M) : ModAlg -> BoolElt, SeqEnum

Returns true if the module M is injective. The function also returns a sequence of multiplicities of the standard injective modules as direct summands of the injective hull of M.

Elementary Operations

m * b : ModAlgElt, AlgBasElt -> ModAlgElt

Given an element b in a basic algebra B and an element m in a module M over B, m * b is the product.

Example `AlgBas_AModules (H83E4)`

We get the dimensions of the radical layers of the group algebra of an extra special group of order 243 over a field of characteristic 3.

> G := ExtraSpecialGroup(3,2);
> G;
Permutation group G acting on a set of cardinality 243
> ff := GF(3);
> A := BasicAlgebra(G,ff);
> A;
Basic algebra of dimension 243 over GF(3)
Number of projective modules: 1
Number of generators: 6
> P := ProjectiveModule(A,1);
> P;
AModule P of dimension 243 over GF(3)
> R := JacobsonRadical(P);
> R;
AModule R of dimension 242 over GF(3)
> while Dimension(R) ne 0 do
>     T := JacobsonRadical(R);
>     print Dimension(R) - Dimension(T);
>     R := T;
> end while;
4
11
20
30
36
39
36
30
20
11
4
1

Example `AlgBas_AModules-2 (H83E5)`

We consider the mod-2 group algebra of an extraspecial group of order 128, and construct the module induced from the trivial module on the subgroups of order 4 generated by the first generator of the group.

> G := ExtraSpecialGroup(2,3);
> G;
Permutation group G acting on a set of cardinality 128
> F := GF(2);
> A := BasicAlgebra(G,F);
> A;
Basic algebra of dimension 128 over GF(2)
Number of projective modules: 1
Number of generators: 8
> A.1;
(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)

Note that cs{A.1} is the unique idempotent (identity element) in the group algebra, whereas cs{A.2} is cs{G.1-1} where cs{G.1} is the first generator of the group.

> A.2;
(0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)
> g := A.1+A.2;

Now we check the order of cs g.

> g^2;
(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)
> g^4;
(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)
> g^4 eq A!1;
true

So cs{g} has order 4.

> P := ProjectiveModule(A,1);
> P;
AModule P of dimension 128 over GF(2)

Note that cs{P} is generated by cs{P.1} which corresponds to the identity element of cs{A} if we think of cs{P} as the algebra cs{A} as a module over itself. Now we create the induced module as the submodule generated by (cs(g) - 1)³, since (cs(g) - 1)⁴ = 0.

> U := sub<P|P.1*A.6>;
> U;
AModule U of dimension 32 over GF(2)

Because the dimension is a quarter of the order of the group we can be sure that we have the right thing by just checking that cs{U} is generated by a cs{g} fixed point.

> U.1*g eq U.1;
true

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Version: V2.14 of Tue Oct 30 14:01:03 EST 2007