[GAP Forum] first example towards GAP, need some explanation

[Dima Pasechnik]

On Sun, Aug 19, 2012 at 02:54:30PM +0800, Jon wrote:
> I know of a 2D irreducible representation to be real for S(4), but the
> 2D irreducible representation returned by GAP is complex. I am
> wondering whether there exists a unique transformation to bring all
> the complex matrices pertinent to GAP to real matrices. Or,
> alternatively, for two representations to be equivalent, should the
> two sets of matrices be related to each other by a single unitrary
> transformation?

there is a unique (up to scalar) Hermitean positive definite
form preserved by the representation rho, obtained as sum_g rho(g)rho(g)*.
One can find then find a transformation which will make rho unitary.
Making rho real (one can actually do even better, making them
rational!) is harder.

IMHO it is easier to construct the rational representations directly
rather than trying to tranform the output of
IrreducibleRepresentations() into a suitable form.  It fact, it's
relatively well-known how to do this.  (it's somehow still outside the
"mainstream" representation theory of finite groups, so even after a
graduate course on the subject people are not likely to know this.)

Young, after whom e.g. Young diagrams are named, already gave explict
formulae for the images of the transpositions (i,i+1) for i=1,...,n-1 in
each irreducible representation rho in his works. A modern explanation
can be found in recent texts, e.g. by Vershik and Okounkouv.
http://www.mat.univie.ac.at/~esiprpr/esi333.pdf
Programming these carefully in GAP would be a good coursework project
for an undegraduate.

HTH,
Dmitrii

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