Standard Constructions

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Standard Constructions for General Modules

Given one or more existing modules, various standard constructions are available to construct new modules.

ChangeRing(M, S) : ModRng, Rng -> ModRng, Map

Given a module M with base ring R, together with a ring S, construct the module N with base ring S obtained by coercing the components of elements of M into N, together with the homomorphism from M to N.

ChangeRing(M, S, f) : ModRng, Rng, Map -> ModRng, Map

Given a module M with base ring R, together with a ring S, and a homomorphism f: R -> S, construct the module N with base ring S obtained by mapping the components of elements of M into N by f, together with the homomorphism from M to N.

DirectSum(M, N) : ModRng, ModRng -> ModRng, Map, Map, Map, Map

Given R-modules M and N, construct the direct sum D of M and N as an R-module. The embedding maps from M into D and from N into D respectively and the projection maps from D onto M and from D onto N respectively are also returned.

DirectSum(Q) : [ ModRng ] -> [ ModRng ], [ Map ], [ Map ]

Given a sequence Q of R-modules, construct the direct sum D of these modules. The embedding maps from each of the elements of Q into D and the projection maps from D onto each of the elements of Q are also returned.

ExteriorSquare(M) : ModTupRng -> ModTupRng

Given an R-module M, construct the R-submodule of M otimes_R M consisting of the skew tensors.

SymmetricSquare(M) : ModTupRng -> ModTupRng

Given an R-module M, construct the R-submodule of M otimes_R M consisting of the symmetric tensors.

TensorProduct(M, N) : ModTupRng, ModTupRng -> ModTupRng

Given R-modules M and N, construct the R-tensor product, M otimes_R N, with diagonal action.

TensorPower(M, k) : ModTupRng, RngIntElt -> ModTupRng

Given a R-module M and an integer k >= 1, construct the k-th tensor power of M.

Example `RMod_Constructions (H42E11)`

We create two 2-dimensional Q-modules M and N, and apply the constructions to them.

> Q := RationalField();
> Q2 := RSpace(Q, 2);
> R := MatrixRing<Q, 2 | [ 0, 1, -1, 0], [ 0, 1, 1, 0 ] >;
> M := RModule(Q2, R);
> M:Maximal;
RModule M of dimension 2 over Rational Field
Generators of acting algebra:


[ 0  1]
[-1  0]


[0 1]
[1 0]
> S := MatrixRing<Q, 2 | [ 0, -1, -1, 0 ], [1, 0, 0, -1 ] >;
> N := RModule(Q2,  S);
> N:Maximal;
RModule M of dimension 2 over Rational Field
Generators of acting algebra:


[ 0 -1]
[-1  0]


[ 1  0]
[ 0 -1]
> P := DirectSum(M, N);
> P:Maximal;
RModule P of dimension 4 over Rational Field
Generators of acting algebra:


[ 0  1  0  0]
[-1  0  0  0]
[ 0  0  0 -1]
[ 0  0 -1  0]


[ 0  1  0  0]
[ 1  0  0  0]
[ 0  0  1  0]
[ 0  0  0 -1]
> Q := ExteriorSquare(M);
> Q:Maximal;
RModule Q of dimension 1 over Rational Field
Generators of acting algebra:


[1]


[-1]
> T := SymmetricSquare(M);
> T:Maximal;
RModule T of dimension 3 over Rational Field
Generators of acting algebra:


[ 0  0  1]
[ 0 -1  0]
[ 1  0  0]


[0 0 1]
[0 1 0]
[1 0 0]

Changing the Coefficient Ring

ExtendField(V, L) : ModTupFld, Fld -> ModTupFld, MapHom

Given a K-vector space V, with K a field and L an extension of K, construct the L-vector space U = V otimes_K L. The function returns

the vector space U; and
the inclusion homomorphism phi : V -> U.

RestrictField(V, L) : ModTupFld, Fld -> ModTupFld, MapHom

Given a K-vector space V, with K a field and L a subfield of K, construct the L-vector space U consisting of those vectors of V having all of their components lying in the subfield L. The function returns

the vector space U; and
the restriction homomorphism phi : V -> U.

VectorSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

KSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

KModule(V, F) : ModTupFld, Fld -> ModTupFld, Map

KMatrixSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

Given an n-dimensional K-vector space V, and a subfield F of K such that K has degree m over F, construct a vector space of dimension mn over the field F. The function returns

the vector space U; and
a mapping phi : V -> U such that each vector (v_1, ..., v_i, ..., v_n) of V is mapped into the vector (u_(11), ..., u_(1n), ..., u_(i1), ..., u_(in), ..., u_(n1), ... u_(nn) ) where (u_(i1), ..., u_(in)) is the field element v_i written as a vector over the subfield F.

Standard Constructions for R[G]-Modules

All of the constructions available for general R-modules may be applied to R[G]-modules. A small number of additional constructions are available only for R[G]-modules.

M ^ T : ModGrp, AlgMatElt -> ModGrp

Given an R[G]-module M of dimension n over the ring R, and a nonsingular n x n matrix T over R, construct the R[G]-module S which corresponds to taking the rows of T as a basis for M.

Dual(M) : ModGrp -> ModGrp

Given an R[G]-module M, construct the the R[G]-module which is the R-dual, Hom_R(M, R), of M.

Induction(M, G) : ModGrp, Grp -> ModGrp

Given an R[H]-module M and a supergroup G of H, construct the R[G]-module obtained by inducing M up to G.

Restriction(M, H) : ModGrp, Grp -> ModGrp

Given a R[G]-module M and a subgroup H of G, form the R[H]-module corresponding to the restriction of M to the subgroup H.

Example `RMod_GModules1 (H42E12)`

Starting with the permutation module M over GF(2) for the Mathieu group M_(24), we construct the restriction N of M to the Sylow 2-subgroup of M_(24).

> M24 := PermutationGroup< 24 |
>         (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24),
>         (2,16,9,6,8)(3,12,13,18,4)(7,17,10,11,22)(14,19,21,20,15),
>         (1,22)(2,11)(3,15)(4,17)(5,9)(6,19)(7,13)(8,20)(10,16)(12,21)
>         (14,18)(23,24)>;
> M := PermutationModule(M24, GaloisField(2));
> M;
GModule M of dimension 24 with base ring GF(2)
> S2 := SylowSubgroup(M24, 2);
> N := Restriction(M, S2);
> N;
GModule N of dimension 24 with base ring GF(2)

Example `RMod_GModules2 (H42E13)`

Starting with the 6-dimensional representation over GF(2) constructed above for A_7, we construct an irreducible 15-dimensional representation for A_7, as the exterior square of the 6-dimensional representation.

> A7 := AlternatingGroup(7);
> V  := VectorSpace(GaloisField(2), 7);
> M  := PermutationModule(A7, V![1,0,1,0,1,0,1]);
> M;
GModule M of dimension 6 with base ring GF(2)
> N := ExteriorSquare(M);
> N: Maximal;
GModule N of dimension 15 with base ring GF(2)
Generators of acting algebra:


[0 1 0 0 0 0 1 0 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 1 1 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 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 1 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 1 1 0 0 0 0 0]
[0 0 1 0 1 1 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 1 0 0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 1 0 1]
[0 0 1 0 0 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 1 1 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 1 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 1 0]
[0 0 0 0 1 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 1]
[0 0 0 0 0 0 0 1 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 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 1 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 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]

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