[GAP Forum] Sqrt for the cyclotomic numbers

[Palcoux Sebastien]

Dear Dima and Forum.
I don't understand how your answer solves my problem, perhaps there is a misunderstanding:

What I want are the unitary matrices representing the elements of the group G for an irreducible representation V.For so, we should conjugate the non-unitary matrices (given by GAP) by the matrix R=S.P with S^{-2} the diagonalization D of the matrix X of the Hermitian positive definite formobtained by the averaging (or in some other way) and P the matrix of the change of basis (into the eigenvectors basis of X).  In this process, we need the find the square root of D, i.e.  the square root of positive cyclotomic numbers.
Is there an other process for doing that without having to compute square root of positive cyclotomic numbers?
Best regards,Sébastien


     Le Mardi 20 janvier 2015 14h29, Dima Pasechnik <dmitrii.pasechnik at cs.ox.ac.uk> a écrit :
   

 On Tue, Jan 20, 2015 at 07:31:56AM +0000, Palcoux Sebastien wrote:
> Dear Alexander and Forum,
> If the cyclotomic number is the square of a cyclotomic number, is there an easy way to find it?
> The number I need are the eigenvalues of the matrix of the unitarized inner product of an irreducible representation of a finite group (see the comment of Paul Garett here: http://math.stackexchange.com/q/1107941/84284). This matrix is positive, I guess its eigenvalues are always cyclotomic (true for the examples I've looked, but I don't know in general), and I hope they are square of cyclotomic. Thanks to these square roots I can compute the unitary matrices for the irreducible representation.

You don't need to take square roots. If H is the Hermitian positive definite form
you obtained by the averaging (or in some other way) then H=LDL*, for 
L a lower-triangular matrix with 1s on the main diagonal, and D is a diagonal matrix.
L and D can be computed without taking square roots (and so they will stay cyclotomic). 
Then conjugating by L gives you the unitary form.

HTH,
Dmitrii


> Remark: a function on GAP computing the unitary irreducible representations seems very natural, so if there is not such a function, this should means that there are problems for computing them in general with GAP, isn't it?
> Best regards,Sebastien Palcoux        
> 
>      Le Mardi 20 janvier 2015 3h13, Alexander Hulpke <hulpke at fastmail.fm> a écrit :
>    
> 
>  Dear Forum,
> 
> > On Jan 19, 2015, at 1/19/15 2:18, Palcoux Sebastien <sebastienpalcoux at yahoo.fr> wrote:
> > 
> > Hi,
> > Is it possible to extend the function Sqrt on the cyclotomic numbers?
> 
> How would you represent this root? In general the square root of a cylotomic is not cyclotomic again. (You could form a formal AlgebraicExtension, but then you lose the irrational cyclotomics for operations.)
> 
> Regards,
> 
>   Alexander Hulpke