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 <hul...@fastmail.fm> a > écrit : > > > Dear Forum, > > > On Jan 19, 2015, at 1/19/15 2:18, Palcoux Sebastien > > <sebastienpalc...@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 _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
