Matrix Group over Symbolic Ring

asked 2018-02-05 05:31:14 +0100

dimahphone
41 ●1 ●2 ●8

updated 2023-05-19 14:29:56 +0100

FrédéricC

5206 ●3 ●43 ●114

I have problem on generate matrix group over symbolic ring.

First, I define

eta=I;
eta2=(1+I)*sqrt(2)/2;

Then, define a generator matrix

T=matrix(SR,4,[eta**(i*j)*eta2/2 for i in range(4) for j in range(4)]);

I try to make a matrix group by

G=MatrixGroup(T);

What I get is only very long computation that does not give any result. Can somebody help me? Thank you very much.

I have checked the order of T, I got 8 since T**8=I where I is identity matrix.

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I've tried instead:

sage: K.<a> = QuadraticField( -1 )
sage: R.<X> = K[]
sage: L.<b> =K.extension( X^2 - 2 )
sage: eta = a
sage: eta2 = (1+a)*b/2
sage: T = matrix(L, 4, [eta**(k*kk)*eta2/2 for k in range(4) for kk in range(4)])
sage: T^8
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: G = MatrixGroup(T)

running into NotImplementedError: Currently, only simple algebraic extensions are implemented in gap. Well... Perhaps gap cannot digest the input in both cases.

Which is the "real need"? Depending on it, the "other way to get the result" may be simpler or more complicated. In the above case one can use block matrices representing $\sqrt{-1}$ and $\sqrt 2$ as commuting $2\times 2$ matrices. If such cases are enough, i will write the code.

dan_fulea ( 2018-02-05 12:23:28 +0100 )edit

Do you mean "real need" is what I need on that code? Sorry if I am wrong understanding your comment.

dimahphone ( 2018-03-07 02:12:59 +0100 )edit

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answered 2018-04-09 06:26:45 +0100

dimahphone
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Case closed, I finally use CyclotomicField(8) and it works.

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@dimahphone -- You can accept your answer by clicking on the accept (tick mark) button, so that this will appear as the accepted answer, and the question will appear as solved in the list of questions.

slelievre ( 2018-04-09 09:25:14 +0100 )edit

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answered 2018-02-05 22:09:37 +0100

slelievre
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Whenever possible, avoid working with Sage's symbolic ring.

Here, you can use the symbolic ring as an intermediate step but move to QQbar.

This will make your code work. So you could start as you did:

sage: eta = I
sage: eta2 = (1+I)*sqrt(2)/2

then instead of defining

sage: T = matrix(SR, 4, [eta**(i*j)*eta2/2 for i in range(4) for j in range(4)])

you could define

sage: T = matrix(QQbar, 4, [eta**(i*j)*eta2/2 for i in range(4) for j in range(4)])

And then things work.

sage: G = MatrixGroup(T)
sage: G
Matrix group over Algebraic Field with 1 generators (
[ 0.3535533905932738? + 0.3535533905932738?*I  0.3535533905932738? + 0.3535533905932738?*I  0.3535533905932738? + 0.3535533905932738?*I  0.3535533905932738? + 0.3535533905932738?*I]
[ 0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I  0.3535533905932738? - 0.3535533905932738?*I]
[ 0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I  0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I]
[ 0.3535533905932738? + 0.3535533905932738?*I  0.3535533905932738? - 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I -0.3535533905932738? + 0.3535533905932738?*I]
)

Calculations in QQbar are exact and efficient.

If you need to see the radical expression of an element in QQbar (if it has one), use .radical_expression().

For example:

sage: T[0,0].radical_expression()
1/2*(-1)^(1/4)

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Thank you for the suggestion. It works. But, I still have problem when I use QQbar. I can not do anything for groups I get. For example, I can not know the element order of the group. By hand, I get that the order is 8. Do you have any suggestion? Thank you very much. Sorry if it disturbs your time.

dimahphone ( 2018-02-06 07:55:16 +0100 )edit

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