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.


1 Answer 1


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 ]
  • $\begingroup$ 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 $[[0,1],[-1,0]] \otimes I_n$; $\endgroup$
    – unknown
    Apr 18, 2021 at 17:38
  • 1
    $\begingroup$ 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. $\endgroup$
    – ahulpke
    Apr 18, 2021 at 21:07

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