ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 08 Jun 2020 14:47:58 -0500Why is the GU(n,q) package the way it is?https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/So I would love to utilize the unitary group feature in Sage; however, it does not seem like Sage defines this group in the standard way that I have seen, namely $n\times n$ matrices over the finite field $\mathbb F_{q^2}$ such that $U^* U = I$, where $U^*$ is the transpose of the matrix in which each entry of $U$ is raised to the $q$ power.
For some weird reason it appears as though sage uses the antitranspose instead of the traditional transpose operation to define unitary matrices, since the following matrix, for example, is included:
\begin{bmatrix}
0 & 1\\\
1 & 1
\end{bmatrix}
Does anyone know why this is and how I can easily rectify my computations so these matrices still preserve the Hermitian form $\langle x,x \rangle = x^*x$?Thu, 04 Jun 2020 15:11:48 -0500https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/Comment by John Palmieri for <p>So I would love to utilize the unitary group feature in Sage; however, it does not seem like Sage defines this group in the standard way that I have seen, namely $n\times n$ matrices over the finite field $\mathbb F_{q^2}$ such that $U^* U = I$, where $U^*$ is the transpose of the matrix in which each entry of $U$ is raised to the $q$ power.</p>
<p>For some weird reason it appears as though sage uses the antitranspose instead of the traditional transpose operation to define unitary matrices, since the following matrix, for example, is included:</p>
<p>\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}</p>
<p>Does anyone know why this is and how I can easily rectify my computations so these matrices still preserve the Hermitian form $\langle x,x \rangle = x^*x$?</p>
https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/?comment=51842#post-id-51842(a) What version of Sage are you using? (b) I should have been clearer about there being no obvious way to change the bilinear form: there is no obvious way to change it in the case when you are defining a finite group, as the example `GU(3,3, invariant_form=[[1,0,0],[0,2,0],[0,0,1]])` illustrates.Mon, 08 Jun 2020 14:47:58 -0500https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/?comment=51842#post-id-51842Comment by nbenj582 for <p>So I would love to utilize the unitary group feature in Sage; however, it does not seem like Sage defines this group in the standard way that I have seen, namely $n\times n$ matrices over the finite field $\mathbb F_{q^2}$ such that $U^* U = I$, where $U^*$ is the transpose of the matrix in which each entry of $U$ is raised to the $q$ power.</p>
<p>For some weird reason it appears as though sage uses the antitranspose instead of the traditional transpose operation to define unitary matrices, since the following matrix, for example, is included:</p>
<p>\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}</p>
<p>Does anyone know why this is and how I can easily rectify my computations so these matrices still preserve the Hermitian form $\langle x,x \rangle = x^*x$?</p>
https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/?comment=51840#post-id-51840So from the documentation I found here: https://doc.sagemath.org/html/en/reference/groups/sage/groups/matrix_gps/unitary.html
It looks like you can change the invariant form, but I tried this and Sage just yells at me saying that 'invariant_form" is an unexpected keyword argument, even if I copy and paste the exact code used there.Mon, 08 Jun 2020 13:47:44 -0500https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/?comment=51840#post-id-51840Comment by John Palmieri for <p>So I would love to utilize the unitary group feature in Sage; however, it does not seem like Sage defines this group in the standard way that I have seen, namely $n\times n$ matrices over the finite field $\mathbb F_{q^2}$ such that $U^* U = I$, where $U^*$ is the transpose of the matrix in which each entry of $U$ is raised to the $q$ power.</p>
<p>For some weird reason it appears as though sage uses the antitranspose instead of the traditional transpose operation to define unitary matrices, since the following matrix, for example, is included:</p>
<p>\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}</p>
<p>Does anyone know why this is and how I can easily rectify my computations so these matrices still preserve the Hermitian form $\langle x,x \rangle = x^*x$?</p>
https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/?comment=51768#post-id-51768These groups are defined using GAP, and GAP defines these groups so that they preserve the bilinear form defined by the matrix `[[0, 1], [1, 0]]`. Once you've defined `G = GU(2, 7)`, you can evaluate `G.invariant_form()` to see this. There is no obvious way (at least to me) to change the bilinear form.Thu, 04 Jun 2020 20:45:03 -0500https://ask.sagemath.org/question/51766/why-is-the-gunq-package-the-way-it-is/?comment=51768#post-id-51768