Dualities and isomorphisms of classical groups
Four of the five families of classical generalised quadrangles come in dual pairs: (i) and ; (ii) and . Both can be demonstrated by the Klein correspondence. Recall from the last post that the Klein correspondence maps a line of represented as the row space of
to the point where
Now consider the symplectic generalised quadrangle defined by the bilinear alternating form
A totally isotropic line must then satisfy
Therefore, the lines of are mapped to points of lying in the hyperplane . Now the quadratic form defining is and is the tangent hyperplane at the projective point , which does not lie in the quadric. Hence the hyperplane is non-degenerate and so we see that maps to points of . That this mapping is bijective follows from noting that the number of lines of is equal to the number of points of (namely, ).
Now we will consider a more difficult situation which reveals that the generalised quadrangles and are also formally dual to one another. Continue reading “The Klein Correspondence II”