# Why is the GU(n,q) 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 $\langle x,x \rangle = x^*x$?

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1

These 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.

( 2020-06-05 03:45:03 +0200 )edit

So from the documentation I found here: https://doc.sagemath.org/html/en/refe...

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.

( 2020-06-08 20:47:44 +0200 )edit

(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.

( 2020-06-08 21:47:58 +0200 )edit