Compact Homogeneous Spaces With Positive Euler Characteristic and Their 'Strange Formulae'

Parabolic Subroot Systems and Their Applications

“…We will normalise (,) to an inner product , so that for the highest rootα we have that α,α = 2. The two inner products are related by (,) = 1 2g , , where g (called the dual Coxeter number) is the eigenvalue of the Casimir element of g C in its adjoint representation (see [11], Proposition 2.1). Choosing a fixed linear functional on E that does not vanish on any of the roots, we can define positive roots + and simple positive roots = {α 1 , .…”