A set of functions are provided for computing with the characters of a group. Full details of these functions may be found in Chapter CHARACTERS OF FINITE GROUPS. For convenience we include here two of the more useful character functions.
Also, functions are provided for computing with the modular representations of a group. Full details of these functions may be found in Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS. For the reader's convenience we include here the functions which may be used to define a R[G]-module for a permutation group.
Construct the table of ordinary irreducible characters for the group G.Al: MonStgElt Default: em "Default"This parameter controls the algorithm used. The string "DS" forces use of the Dixon-Schneider algorithm. The string "IR" forces the use of Unger's induction/reduction algorithm [Ung06a]. The "Default" algorithm is to use Dixon-Schneider for groups of order ≤5000 and Unger's algorithm for larger groups. This may change in future.DSSizeLimit: RngIntElt Default: 0When the default algorithm is selected, a positive value n for DSSizeLimit means that before using Unger's algorithm, the full character space is split by some passes of Dixon-Schneider, restricted to using class matrices corresponding to conjugacy classes with size at most n.
Given a group G represented as a permutation group, construct the character of G afforded by the defining permutation representation of G.
Given a group G and some subgroup H of G, construct the ordinary character of G afforded by the permutation representation of G given by the action of G on the coset space of the subgroup H in G.
Let G be a group defined on r generators and let S be a subalgebra of the matrix algebra Mn(R), also defined by r non-singular matrices. It is assumed that the mapping from G to S defined by φ(G.i) -> S.i, for i = 1, ..., r, is a group homomorphism. Let M be the natural module for the matrix algebra S. The function GModule gives M the structure of an S[G]-module, where the action of the i-th generator of G on M is given by the i-th generator of S.
Given a finite group G, a normal subgroup A of G and a normal subgroup B of A such that the section A/B is elementary abelian of order pn, create the K[G]-module M corresponding to the action of G on A/B, where K is the field GF(p). If B is trivial, it may be omitted. The function returns
- (a)
- the module M; and
- (b)
- the homomorphism φ : A/B -> M.
Given a finite group G and a ring R, create the R[G]-module for G corresponding to the permutation action of G on the cosets of H.
Given a finite permutation group G and a ring R, create the natural permutation module for G over R.
> G := PermutationGroup<24 |
> [ 3, 4, 1, 2,23,24, 7, 8, 9,10,12,11,14,13,16,15,18,17,22,21,
> 20,19, 5, 6 ],
> [ 7, 8,11,12,13,14,22,21,20,19,15,16,17,18, 6, 5, 4, 3, 1, 2,23,
> 24, 9,10 ] >;
> N := sub<G |
> [ 24, 23, 6, 5, 4, 3, 10, 9, 8, 7, 14, 13, 12, 11, 18, 17, 16, 15, 22, 21,
> 20, 19, 2, 1 ],
> [ 23, 24, 5, 6, 3, 4, 8, 7, 10, 9, 12, 11, 14, 13, 15, 16, 17, 18, 19, 20,
> 21, 22, 1, 2 ],
> [ 2, 1, 4, 3, 6, 5, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 15, 16, 21, 22, 19,
> 20, 24, 23 ]>;
> #N;
8
> IsNormal(G, N);
true
> IsElementaryAbelian(N);
true
> M, f := GModule(G, N);
> SM := Submodules(M);
> #SM;
4
> refined := [ x @@ f : x in SM ];
> forall{x : x in refined | IsNormal(G, x) };
true;
> [ #x : x in refined];
[ 1, 2, 4, 8 ]
The original elementary abelian normal subgroup of order 8 is the top
of a chain of normal subgroups of length 3.