See examples in FGR.mw
To use this set of commands for a given group, create the record
GR::Record(
name::string group name (all the strings here and below may be empty);namegen::list(string) names of generators;gen::list(g) generators (unit element is not needed);GF::proc(gen) function generating the group, elements ordering must not depend on generators;nameC::list(string) names of conjugacy classes;GFC::proc(gen) function generating representatives of conjugacy classes;NC::list(posint) list of the numbers of elements in conjugacy classes;nameR::list(string) names of irreducible representations;genR::list(list(Matrix)) generators of irreducible representations corresponding to gen.FGR_ChTb(GR,{fulloutput})::Matrix builds character table of a given matrix group,
if fulloutput returns also labels for representations and conjugacy classes.FGR_RepU(R0,R)::list(Matrix) for any representation R0
returns the list of matrices U of unitary transformation
to irreducible representation R, so that U+.R0.U=R.
Note that R0 and R must have the same ordering.
See also FGR_RepU2 in FiniteGroups.ini.
FGR_RepDec(R,GR,/IRs,modgen,{aslist})::list([string,list(Matrix)] or [string,Matrix,dimension])
decomposes representation R into irreducible representations of GR.
The result is a list of names and matrices of unitary transformation to irreducible representations.
The IRs limits decomposition to IRs,
the modgen="real" modifies generators to make them real-valued.FGR_Ch(R,GR)::list(nonnegint) returns the characters of elements
in R corresponding to conjugacy classes representatives.FGR_RepDecCh(chi::list,ChTb::Matrix,NC::list)::list(nonnegint)
short form of the decomposition of any representation given by its characters chi
into irreducible representation of a group given by its character table ChTb
and list NC of the numbers of elements in conjugacy classes.
The result is a list of irreducible representations numbers.FGR_indexC(GR)::list(nonnegint) returns the list of indexes
of conjugacy classes representatives generated by GR:-GFC.FGR_dimR(GR)::list(nonegint) returns the list of dimensions of irreducible representations.