Given one or more existing modules, various standard constructions are available to construct new modules.
In this collection of functions will be found utilities for changing the base ring of the module. Note that several of the functions for rewriting over a minimal field are restricted to rings K[G] where K is a finite field.
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.
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.
The functions in this section currently only apply to K[G]-modules defined over a finite field K.
Given a K[G]-module M, where K is a finite field, return true if M can be realised over a proper subfield F of K. The equivalent F[G]-module is also returned. The Glasby-Howlett algorithm is used to determine the smallest field over which M can be realised.
Let M be a K[G]-module, where K is a finite field of characteristic p, and let F be a finite field also of characteristic p. If it is possible to realise M over the subfield F of K, return true and the equivalent F[G]-module.
Given a module of dimension d over a finite field E having degree e and a subfield F of E having degree f, write the action of M as d * e/f by d * e/f matrices over F and return the module and the isomorphism.
Let M be a K[G]-module, where K is a finite field of characteristic p, and let F be a finite field also of characteristic p. This function returns the module obtained by writing M over the smallest possible field containing F subject to the condition that the dimension of M does not increase. The Glasby-Howlett algorithm is used to determine the smallest field over which M can be realised.
Let Q be a sequence of K[G]-modules, where K is a finite field of characteristic p, and let F be a finite field also of characteristic p. This function returns the sequence of modules obtained by writing each module M of Q over the smallest possible field containing F subject to the condition that the dimension of M does not increase. Thus, the effect of the function is to apply the function AbsoluteModuleOverMinimalField to each module of Q. The Glasby-Howlett algorithm is used to determine the smallest field over which the modules M of Q can be realised.
Let M be a K[G]-module of dimension d, where K is a finite field of characteristic p, and let F be a subfield of K of index n. This function returns the F[G]-module N obtained by writing the action of M as dn x dn matrices over F.
Let Q be a sequence of K[G]-modules, where K is a finite field of characteristic p, and let F be a subfield of K of index n. This function returns the sequence R of F[G]-modules obtained by applying the function ModuleOverSmallerField to each term of Q. That is, each term N of R is formed by writing the action of the corresponding term of Q as dn x dn matrices over F.
Given A-modules M and N, change their base fields to K, where K is the smallest field containing the base fields of M and N.
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.
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.
Let M and N be two K[G] modules. This function constructs the tensor product, M otimes_A N, with diagonal action.
Given a K[G]-module M and an integer k >= 1, construct the n-th tensor power of M.
Given a K[G]-module M, construct the A-submodule of M otimes_A M consisting of the skew tensors.
Given a K[G]-module M, construct the A-submodule of M otimes_A M consisting of the symmetric tensors.
Given an K[G]-module M, construct the K[G]-module which is the K-dual, Hom_K(M, K), of M.
Given a K[H]-module M and a supergroup G of H, construct the K[G]-module obtained by inducing M up to G.
Given a representation R of a subgroup of G, construct the representation of G obtained by inducing R up to G.
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.
> 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.
Given an K[G]-module M, construct the largest submodule of M on which G acts trivially, i.e. the fixed-point space of M.
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.[Next][Prev] [Right] [Left] [Up] [Index] [Root]