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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 K[G]-module for a matrix group.
The functions described in this section apply only to finite groups for
which a base and strong generating set may be constructed.
A sequence containing the linear characters for the group G.
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 and a subgroup H of G, construct the ordinary
character afforded by the representation of G given by its action on
the coset space of the subgroup H.
The natural R[G]-module for the matrix group G.
Let A be a matrix ring defined over the ring R and let G be a
finite group defined on m generators. Let M denote the underlying
module of A. Suppose there is a one-to-one correspondence between
the generators of G and the generators [ A1, ..., Am ] of A.
The function GModule creates the R[G]-module corresponding to an
action of G on M defined by A, where the action of the i-th
generator of G on M is given by Ai.
Let A be a matrix ring defined over the ring R and let G be a
finite group defined on m generators. Let M denote the underlying
module of A. Given a sequence Q of m elements of A, the function
GModule creates the R[G]-module corresponding to an action of
G on M defined by Q, where the action of the i-th
generator of G on M is given by Q[i].
Let A and B be normal subgroups of G such that B is contained
in A. Further, assume that A/B is elementary abelian of order pn,
p a prime. Let K denote the field of p elements. This function
constructs a K[G]-module corresponding to the action of the group G
on the elementary abelian section A/B of G. The map from A to the
K[G]-module's underlying vector space is also returned.
The permutation module for the matrix group G over the ring R defined by its
action on the cosets of the subgroup H.
Given a matrix group G and a submodule S of its natural module,
return an invertible matrix with topmost rows a basis for S.
Conjugating by the inverse of this matrix puts the generators of G
into a block form that exhibits their action on S and the quotient module.
We use the module machinery to refine an elementary abelian
normal subgroup by finding a normal subgroup contained in it.
> G := MatrixGroup<4, IntegerRing(4) |
> [ 3, 3, 1, 3, 0, 2, 2, 3, 3, 0, 1, 3, 3, 2, 2, 1 ],
> [ 2, 2, 3, 3, 0, 3, 1, 1, 3, 0, 1, 1, 2, 0, 1, 2 ] >;
> #G;
660602880
> H := pCore(G, 2);
> FactoredOrder(H);
[ <2, 15> ]
> IsElementaryAbelian(H);
true
> M, f := GModule(G, H, sub<H|>);
> SM := Submodules(M);
> #SM;
3
One of these submodules is 0, one is all M,
we are interested in the one in the middle. Note that the
result returned by Submodules is sorted by dimension.
> N := SM[2] @@ f;
> N;
MatrixGroup(4, IntegerRing(4))
Generators:
[3 0 0 0]
[0 3 0 0]
[0 0 3 0]
[0 0 0 3]
We have found N, a normal subgroup of G, contained
in the 2-core, with order 2.
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