Chapter CHARACTERS OF FINITE GROUPS on characters describes many functions for computing with partial character tables or individual characters.
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 [Con90b].The second form requires z to be a central element of G and p to be a prime or zero. The sequence returned enumerates the number of absolutely irreducible characters of G in characteristic p, lying over some faithful linear character of < z >.
Given a finite p-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 [Sla86].
Given a finite p-group G, return the sequence [C0, C1, ... ], where Ci is the number of irreducible characters of G having degree pi. For details of the algorithm see [Sla86].
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 [Ung06]. 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: 10^4When 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 finite p-group G, return the character table of G. The algorithm is due to Conlon, as described in [Con90a].
The G-module for the action of G on the vector space defined by the matrix ring M.
A KG-module M corresponding to the action of the group G on the elementary abelian subgroup A of G is constructed. The map from A to the vector space underlying M is also returned.
A KG-module M corresponding to the action of the group G on the elementary abelian section A/B of G is constructed. The map from A to the vector space underlying M is also returned.
Compute the absolutely irreducible representations of the group G over appropriate extensions or sub-fields of the given field k. The representations returned are inequivalent and consist of all distinct representations, subject to the conditions imposed. The field k may be a finite field, the rationals or a cyclotomic field. In the case when k is a finite field, the Glasby-Howlett algorithm is used to determine the minimal field over which a representation may be realised. If k has characteristic 0, the field over which a representation is realised may not be minimal.The representations are found using Schur's method of climbing the composition series for G defined by the pc-presentation. If the argument i is given then the algorithm will calculate only representations of the ith subgroup of the composition series.
The "Representations" function returns a list of homomorphisms ρ : G to GL(n, K), where K is a field compatible with k. The "Modules" version returns an equivalent list of G-modules.
Process: BoolElt Default: trueIf the parameter Process is set true then the list is a list of pairs comprising an integer and a representation. This list or any sublist of it is a suitable value for the argument L in the last versions of the function, and in this case only the representations in L will be extended up the series. This allows the user to inspect the representations produced along the way and cull any that are uninteresting.GaloisAction: MonStgElt Default: em "Yes"Possible values are "Yes", "No" and "Relative". The default is "Yes" for intermediate levels and "No" for the whole group. The value "Yes" means that it only lists one representation from each orbit of the action of the absolute Galois group Gal(K/hbox(primefield(K))). Setting this parameter to "No" turns this reduction off (thus listing all inequivalent representations), while setting it to "Relative" uses the group Gal(K/k).MaxDimension: RngIntElt Default:Restrict the representations to those of dimension ≤ MaxDimension. The default is no restriction.ExactDimension: SetEnum Default:If ExactDimension is assigned a set S of positive integers, attention is restricted to representations having dimensions lying in the set S. The default is equivalent to taking the set of all positive integers.If both MaxDimension and ExactDimension are assigned values, then representations having dimensions that are either bounded by MaxDimension or contained in ExactDimension are produced.
Compute irreducible representations of G over the given field k. All arguments and parameters are as for the absolutely irreducible case.The computation proceeds by first computing the absolutely irreducible representations subject to the given parameters, then rewriting over the field k, with a consequent change of dimension of the representation.
> G := DihedralGroup(GrpPC, 10); > FactoredOrder(G); [ <2, 2>, <5, 1> ]
First some modular representations with characteristic 2.
> r := IrreducibleModulesSchur(G, GF(2));
> r;
[*
GModule of dimension 1 over GF(2),
GModule of dimension 4 over GF(2)
*]
> r := AbsolutelyIrreducibleModulesSchur(G, GF(2));
> r;
[*
GModule of dimension 1 over GF(2),
GModule of dimension 2 over GF(2^2),
GModule of dimension 2 over GF(2^2)
*]
> r := AbsolutelyIrreducibleModulesSchur(G, GF(2) : GaloisAction:="Yes");
> r;
[*
GModule of dimension 1 over GF(2),
GModule of dimension 2 over GF(2^2)
*]
The irreducible representation of dimension 4 is not absolutely irreducible,
as over GF(4) it splits into two Galois-equivalent representations.
Getting irreducible representations over the complex field presents no problem, despite not being able to use the complex field as an argument to the function call. We could specify the field to be the cyclotomic field with degree equal to Exponent(G), but it is preferable to ask for absolutely irreducible representations over the rationals.
> r := AbsolutelyIrreducibleRepresentationsSchur(G, Rationals());
> r;
[*
Mapping from: GrpPC: G to GL(1, RationalField()),
Mapping from: GrpPC: G to GL(1, RationalField()),
Mapping from: GrpPC: G to GL(1, RationalField()),
Mapping from: GrpPC: G to GL(1, RationalField()),
Mapping from: GrpPC: G to GL(2, CyclotomicField(5)),
Mapping from: GrpPC: G to GL(2, CyclotomicField(5)),
Mapping from: GrpPC: G to GL(2, CyclotomicField(5)),
Mapping from: GrpPC: G to GL(2, CyclotomicField(5))
*]
> r[6](G.2);
[zeta_5^3 0]
[ 0 zeta_5^2]