[GAP Forum] testing representations

Frank Lübeck frank.luebeck at math.rwth-aachen.de
Thu May 29 10:59:06 BST 2008

Dear Vahid, dear Forum,

On Fri, May 23, 2008 at 02:32:57PM -0700, Vahid Dabbaghian wrote:
> Suppose a character chi and an ordinary representation R of G, both of
> degree d, are given. One way to test that R is a representation affording
> chi is to compute the trace of R(x) for representatives x of conjugacy
> classes. In the case that d and the entries of R(x) are large, finding R(x)
> and computing the trace is an expensive task. Do you have any suggestion for
> a faster way to do this test? 

. . . it depends!

If you have exactly the information as stated above, I think you cannot do
much better. But in practice you probably have additional information. Here
are a few thoughts what could make the task faster in certain cases.

If you have an R(x) for which it is not to bad to compute the eigenvalues,
and if you know the power map for x (i.e., the classes which contain the
powers x^i), then you can quickly compute the eigenvalues (and traces) of
all R(x^i).

If it is likely that your chi is not the character of R or a algebraic conjugate
of it, it may be sufficient to compute the traces of some random elements 
in the image of R to find a trace which doesn't occur as value of chi. 

If you know that chi is irreducible and you know all irreducible characters 
of degree d of your group (or, if you know all characters of degree d from
the character table), it may be sufficient to compute just a few specific or 
random R(x) to identify the character of R.

Best regards,


///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64,  ///
\\\                    52062 Aachen, Germany                          \\\
///  E-mail: Frank.Luebeck at Math.RWTH-Aachen.De                        ///
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