### 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 ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### 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 ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### 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 ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### 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. ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### 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 ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### 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 ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum