Morse-Bott functions on orthogonal groups

TL;DR

This paper studies Morse-Bott functions on orthogonal groups, analyzing critical points, indices, and Betti numbers, providing new computational methods and clarifications of existing results.

Contribution

It offers a detailed analysis of trace functions on orthogonal groups, including new Morse-theoretic computations of Betti numbers and simplified treatments of existing results.

Findings

Critical loci of quadratic trace functions are described.

Indices are determined via perfect fillings of tables.

New Morse-theoretic computation of mod 2 Betti numbers of SO(n).

Abstract

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$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B$. We give a simplified treatment of T. Frankel's analysis of the linear trace function on $SO(n)$, as well as a combinatorial explanation of the relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysis. 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 $SO(n)$.