### Re: [GAP Forum] Sqrt for the cyclotomic numbers

On Tue, Jan 20, 2015 at 02:05:30PM +, Dima Pasechnik wrote:
Dear Sebastien,

Once again, your X can be written as X=L*DL, with D diagonal and real. The
group L^-1 G L preserves the Hermitian form x*Dx. In particular any g in this
group satisfies g*Dg=D.

As it acts irreducibly, D is a scalar matrix, thus g  is unitary.
Sorry, this last claim is wrong: to get a unitary g, you will need
to take D^(1/2) g D^(-1/2). I suppose this is still easier
to compute than taking square roots during the diagonalisation of X.

Indeed, for computing D cyclotomics might not suffice, but we do not need D
explicitly.
here I meant computing D^(1/2), certainly, not just D.

I shall never again write to Form from a mobile phone. :-)

Dima

Dima

On 20 Jan 2015 10:07, Palcoux Sebastien sebastienpalc...@yahoo.fr wrote:

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 form
obtained 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.pasech...@cs.ox.ac.uk a écrit :

On Tue, Jan 20, 2015 at 07:31:56AM +, 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

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### Re: [GAP Forum] Sqrt for the cyclotomic numbers

On Tue, Jan 20, 2015 at 07:31:56AM +, 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

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### Re: [GAP Forum] Sqrt for the cyclotomic numbers

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.pasech...@cs.ox.ac.uk a écrit :

On Tue, Jan 20, 2015 at 07:31:56AM +, 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

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### Re: [GAP Forum] Sqrt for the cyclotomic numbers

Le Mardi 20 janvier 2015 2h48, Palcoux Sebastien sebastienpalc...@yahoo.fr
a écrit :

Hi,
Is it possible to extend the function Sqrt on the cyclotomic numbers?
See below what's happen if we try, for example, with the number E(5) :
gap Sqrt(E(5));Error, no method found! For debugging hints type ?Recovery
from NoMethodFoundError, no 1st choice method found for `Sqrt' on 1 arguments
called fromfunction( arguments ) called from read-eval-loopEntering break
read-eval-print loop ...you can 'quit;' to quit to outer loop, oryou can
'return;' to continuebrk

Sqrt(E(5)) is +/- E(5)^3, and more generally Sqrt(E(p)) is +/- E(p)^((p+1)/2)
for odd p.

gap (E(5)^3)^2;
E(5)

However, in the general the square root of a cyclotomic number is not a
cyclotomic number, for example Sqrt(2) is a cyclotomic number, but not
Sqrt(Sqrt(2)).

Cheers,
Bill.

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### Re: [GAP Forum] Sqrt for the cyclotomic numbers

Dear Sebastien,

Once again, your X can be written as X=L*DL, with D diagonal and real. The
group L^-1 G L preserves the Hermitian form x*Dx. In particular any g in this
group satisfies g*Dg=D. As it acts irreducibly, D is a scalar matrix, thus g is
unitary.

Indeed, for computing D cyclotomics might not suffice, but we do not need D
explicitly.

Dima

On 20 Jan 2015 10:07, Palcoux Sebastien sebastienpalc...@yahoo.fr wrote:

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 form
obtained 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.pasech...@cs.ox.ac.uk a écrit :

On Tue, Jan 20, 2015 at 07:31:56AM +, 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

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### Re: [GAP Forum] Sqrt for the cyclotomic numbers

Hi,
In fact, I need Sqrt only for the positive cyclotomic numbers.
Sebastien

Le Mardi 20 janvier 2015 2h48, Palcoux Sebastien
sebastienpalc...@yahoo.fr a écrit :

Hi,
Is it possible to extend the function Sqrt on the cyclotomic numbers?
See below what's happen if we try, for example, with the number E(5) :
gap Sqrt(E(5));Error, no method found! For debugging hints type ?Recovery from
NoMethodFoundError, no 1st choice method found for `Sqrt' on 1 arguments called
fromfunction( arguments ) called from read-eval-loopEntering break
read-eval-print loop ...you can 'quit;' to quit to outer loop, oryou can
'return;' to continuebrk
Best regards,Sebastien Palcoux

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