This is a continuation of my last post on this subject. As Gordon remarked in one of his posts, you may need to refresh your browser if some of the embedded gifs do not appear as they should.

### 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, ).

Hence .

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”