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 
       
M=Mat(GF(3),4,4);M 
       
J=Matrix(GF(3),[[0,0,-1,0],[0,0,0,-1],[1,0,0,0],[0,1,0,0]]);J 
       
g=G.random_element();g 
       

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) 
       

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

M(g)*J*M(g).transpose() 
       

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.