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For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector bundles on projective space), come with a standard basis of `roots' (elements of square $2$), and the Gram matrix with respect to this basis of roots has largish positive integers off of the diagonal. I'm interested in the group generated by the reflections in these roots, which I believe is free Coxeter. If the off-diagonal entries were very negative, then standard Coxeter combinatorics would imply this, but that is not what is given in this part of nature.

My question is if other people have encountered similar situations in their neck of the woods and if there are techniques for dealing with such a situation.

Searching the literature for a long time only turned up two papers about 'higher rank Coxeter groups' of signature $(p,q)$. While these papers didn't exactly help, I imagine that knowing more about higher rank Coxeter groups would be indirectly useful, so any pointers on such groups are also welcome.

EDIT: I should add that I know how to deal with the case $n=1, (2,1)$, so am more interested in $n \geq 2$.

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