On 30/01/2016 16:57, Fred Lunnon wrote:
I have been struggling to recall a rather surprising theorem, encountered in passing while searching for something unrelated, concerning the structure of quadratic spaces for which the Witt index exceeds 2 . Not only can I not find the theorem; I can't even locate a definition of the Witt index --- that is |p - q| + r , where the (possibly degenerate) quadratic form signature involves p positive, q negative, r absent squares.
The definition at http://math.uga.edu/~pete/quadraticforms.pdf (which was the first hit I got for <<"witt index" "quadratic form">>) seems to imply a different value for the Witt index. Theorem 7.6 there says, I'm understanding it right, that a quadratic space has an orthogonal decomposition as R+D+H where - R is the "radical" and consists of vectors orthogonal to everything - D is anisotropic, i.e. nothing in it has q(v,v)=0 - H is a sum of "hyperbolic planes" (spanned by vectors v,w with q(v,v)=1, q(w,w)=-1, q(v,w)=0) and the Witt index is the number of hyperbolic planes. So, e.g., if the space is R^n and the quadratic form is the usual Euclidean inner product then in your notation we have p=n, q=0, r=0 so "your" Witt index is n; but the decomposition here is 0 + whole_space + 0 and the Witt index is 0. What am I missing? (Almost certainly something obvious. I don't know anything about this stuff.) -- g