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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: 0
When 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.
We refine an elementary abelian normal subgroup of
a permutation group to a sequence of normal subgroups.
> 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.
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