# symplectic groups in GAP

For the symplectic groups in GAP : grp:=SymplecticGroup(n,q), what is the underlying metric or bilinear form ($$M$$)? For example if I use A:=Random(grp) to get a random symplectic matrix this matrix should satisfy $$A^\dagger A=1$$ where $$A^\dagger=M A^t M^{-1}$$; what is $$M$$? I tried $$[[0,-I],[I,0]]$$ but that didn't work.

As explained in the manual (https://www.gap-system.org/Manuals/doc/ref/chap50.html#X8674AAA578FE4AEE), the form is stored in the InvariantBilinearForm attribute (as entry .matrix):

gap> grp:=SP(4,5);;
gap> M:=InvariantBilinearForm(grp).matrix;;
gap> Display(M);
. . . 1
. . 1 .
. 4 . .
4 . . .
gap> List(GeneratorsOfGroup(grp),x->TransposedMat(x)*M*x=M);
[ true, true ]

• Thanks; that answers the question. Curious about the choice of this particular form; it's different from the more "traditional" : $I_n \otimes [[0,1],[-1,0]]$ or $[[0,1],[-1,0]] \otimes I_n$; Apr 18, 2021 at 17:38
• It is often arguable what choice is the standard''. However here the choice of form stems from the construction of the groups as Chevalley groups (maths.usyd.edu.au/u/don/papers/genAC.pdf) -- the choice is made to write matrix generators nicely, and the form then is taken. Apr 18, 2021 at 21:07