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A set of functions are provided for computing with the characters and
representations of a group. A full account of the character functions
may be found in Chapter CHARACTERS OF FINITE GROUPS. Full details of the functions for constructing
and analyzing representations may be found in Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS.
For the reader's convenience we include here a
description of the basic functions for creating characters and
representations.
Some functions described in this section may be missing or may have slightly
different calling sequences for some categories of groups.
For a complete description of the features available for a special category
of groups, we refer to the chapter devoted to that category.
Given a finite pc-group G, return the sequence
[< d1, c1 >, < d2, c2 >, ... ],
where ci is the number of irreducible characters of G having degree
di. For details of the algorithm see Conlon [Con90].
Construct the table of irreducible characters for the group G.
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.
For a finite group G computes the Burnside cokernel C(G),
the quotient of the rational representation ring RQ(G) by the image of the
Burnside ring (virtual permutation representations).
Returns C(G) as a finite abelian group, map RQ(G)to C(G),
and a list of irreducible rational characters with non-trivial
class in C(G).
(The implementation follows [AD14], and does not compute the subgroup
lattice.)
The first example of a finite group with C(G)≠1 is due to Serre,
G=C 3 x Q 8. In fact, there are two groups of order 24 with C(G)≠1 and
no smaller examples:
> [GroupName(G): G in SmallGroups(n), n in [1..24] | #BurnsideCokernel(G) ne 1];
[ C3:C8, C3*Q8 ]
We describe the main functions for creating K[G]-modules for finite
groups. The machinery for working with these modules is described in
Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS.
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, extends to 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.
The permutation module for the group M 10 over GF(2)
may be created as follows:
> m10 := PermutationGroup< 10 | (1, 3, 9, 10, 2, 8, 7, 6, 4, 5),
> (1, 7)(2, 4, 3, 6, 8, 10, 9, 5) >;
> p := PermutationModule(m10, GF(2));
> p : Maximal;
GModule p of dimension 10 over GF(2)
Generators of acting algebra:
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0]
The group G defined below is the split extension of an
elementary abelian group E of order 16 by Alt(6). After setting
up the group, we construct the module M for G corresponding to
its action on E.
> G := PermutationGroup< 16 |
> (1, 15, 7, 5, 12)(2, 9, 13, 14, 8)(3, 6, 10, 11, 4),
> (1, 4, 5)(2, 8, 10)(3, 12, 15)(6, 13, 11)(7, 9, 14),
> (1, 16)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11)(12, 13)(14, 15) >;
> CS := ChiefSeries(G);
> [ Order(H) : H in CS ];
[ 5760, 16, 1 ]
> M := GModule(G, CS[2]);
> M:Maximal;
GModule M of dimension 4 over GF(2)
Generators of acting algebra:
[0 1 0 0]
[0 1 1 0]
[0 0 1 1]
[1 0 0 1]
[0 0 1 0]
[0 0 0 1]
[1 0 1 0]
[0 1 0 1]
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
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