67.51 SpinorNorm

SpinorNorm( form, mat )

computes the spinor norm of mat with respect to the symmetric bilinear form.

The underlying field must have odd characteristic.

    gap> z  := GF(9).root;;
    gap> m1 := [[0,1,0,0,0,0,0,0,0],[1,2,2,0,0,0,0,0,0],
    >  [0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],
    >  [0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],
    >  [0,2,1,0,0,0,0,0,0]]*z^0;;
    gap> m2 := [[z,0,0,0,0,0,0,0,0],[0,z^7,0,0,0,0,0,0,0],
    >  [0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],
    >  [0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],
    >  [0,0,0,0,0,0,0,0,1]]*z^0;;
    gap> form := IdentityMat( 9, GF(9) );;
    gap> form{[1,2]}{[1,2]} := [[0,2],[2,0]] * z^0;;
    gap> m1 * form * TransposedMat(m1) = form;
    true
    gap> m2 * form * TransposedMat(m2) = form; 
    true
    gap> SpinorNorm( form, m1 );
    Z(3)^0
    gap> SpinorNorm( form, m2 );
    Z(3^2)^5