Morse-Bott functions on orthogonal groups

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups O(n). We describe the critical loci of the quadratic trace function Tr(AXBXT) and determine their indices via perfect fillings of tables. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod 2 Betti numbers of SO(n).