# What_the_hell?

## 3145 days ago by jimlb

Ok, for some reason I am missing something here.. My brain is a little shot, but still...

G=Sp(4,GF(3));G
 $\newcommand{\Bold}{\mathbf{#1}}\text{Sp}_{4}(\Bold{F}_{3})$
M=Mat(GF(3),4,4);M
 $\newcommand{\Bold}{\mathbf{#1}}\mathrm{Mat}_{4\times 4}(\Bold{F}_{3})$
J=Matrix(GF(3),[[0,0,-1,0],[0,0,0,-1],[1,0,0,0],[0,1,0,0]]);J
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)$
g=G.random_element();g
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 2 & 1 & 2 & 0 \\ 0 & 2 & 2 & 1 \\ 0 & 0 & 2 & 2 \\ 2 & 1 & 2 & 2 \end{array}\right)$

So $g$ is a random element in the symplectic group, however when I go to check it is actually symplectic:

M(g).transpose()*J*M(g)
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 1 & 0 & 1 \\ 2 & 0 & 2 & 1 \\ 0 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \end{array}\right)$

And just to make sure the transpose is on the correct side:

M(g)*J*M(g).transpose()
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 1 & 0 & 1 \\ 2 & 0 & 2 & 1 \\ 0 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \end{array}\right)$

Neither of those is $J$, so it seems SAGE (which uses GAP) must use some other form to define the symplectic group or I am missing something obvious.  I tried $J = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$ and that didn't work either.  Do you see what I'm messing up? It must be something obvious.