Standard Constructions

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

Subsections

• Changing the Coefficient Ring
• Direct Sum
• Tensor Products of K[G]-Modules
• Induction and Restriction
• The Fixed-point Space of a Module
• Changing Basis

Changing the Coefficient Ring

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

Given an A-module M with base ring R, together with a ring S, such that there is a natural homomorphism from R to S, construct the module N with base ring S where N is obtained from M by coercing the components of the vectors of M into N. The corresponding homomorphism from M to N is returned as a second value.

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, where N is obtained from M by applying f to the components of the vectors of M. The corresponding homomorphism from M to N is returned as a second value.

Direct Sum

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.

Tensor Products of K[G]-Modules

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

Let M and N be two A modules, where A = K[G]. This function constructs the tensor product, M otimes_A N, with diagonal action.

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

Given a K[G]-module M and an integer k >= 1, construct the n-th tensor power of M.

ExteriorSquare(M) : ModTupRng -> ModTupRng

Given an A module M, where A = K[G], construct the A-submodule of M otimes_A M consisting of the skew tensors.

SymmetricSquare(M) : ModTupRng -> ModTupRng

Given an A module M, where A = K[G], construct the A-submodule of M otimes_A M consisting of the symmetric tensors.

Induction and Restriction

Dual(M) : ModGrp -> ModGrp

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

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

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

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

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

The Fixed-point Space of a Module

Fix(M): Mod -> Mod

Given an A-module M, construct the largest submodule of M on which G acts trivially, i.e. the fixed-point space of M.

Example ModAlg_GModules1 (H76E12)

> SetSeed(1);
> G := PermutationGroup< 22 |
>         (1,2,4,8,16,9,18,13,3,6,12)(5,10,20,17,11,22,21,19,15,7,14),
>         (1,18,4,2,6)(5,21,20,10,7)(8,16,13,9,12)(11,19,22,14,17),
>         (1,18,2,4)(3,15)(5,9)(7,16,21,8)(10,12,20,13)(11,17,22,14) >;
> M := PermutationModule(G, GaloisField(2));
> M;
GModule M of dimension 22 with base ring GF(2)
> CM := Constituents(M);
> CM;
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 10 over GF(2),
    GModule of dimension 10 over GF(2)
]

We restrict the module M to the stabilizer of a point in M_(22) and then induce back up, a constituent of the restriction.

> L34 := Stabilizer(G, 1);
> N := Restriction(M, L34);
> N;
GModule N of dimension 22 with base ring GF(2)
> CN := Constituents(N);
> CN;
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 9 over GF(2),
    GModule of dimension 9 over GF(2)
]
> Ind1 := Induction(CN[1], G);
> Ind1;
GModule Ind1 of dimension 22 over GF(2)
> Constituents(Ind1);
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 10 over GF(2),
    GModule of dimension 10 over GF(2)
]
> Ind2 := Induction(CN[2], G);
> Ind2;
GModule Ind2 of dimension 198 over GF(2)
> Constituents(Ind2);         
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 10 over GF(2),
    GModule of dimension 10 over GF(2),
    GModule of dimension 34 over GF(2),
    GModule of dimension 98 over GF(2)
]

Thus, inducing up the 1-dimensional constituent of N gives us irreducible modules for G having the same dimensions as those appearing as constituents of M. However, inducing up the 9-dimensional module gives us irreducible modules of new dimensions: 34 and 98. Hence starting out with only the permutation module for M_(22) over GF(2), we have found 5 irreducible modules for the group.

Changing Basis

M ^ T : ModGrp, AlgMatElt -> ModGrp

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