The index of equidimensional flag manifolds

TL;DR

This paper computes the Fadell-Husseini index for a specific class of flag manifolds with a cyclic group action, extending previous work on Grassmannians and impacting geometric analysis of convex bodies.

Contribution

It provides a complete calculation of the index for equidimensional flag manifolds with cyclic symmetry, offering new insights into their topological and geometric properties.

Findings

Complete calculation of the Fadell-Husseini index for the flag manifold

Extension of previous Grassmannian index computations to flag manifolds

Implications for p-fold orthogonal shadows of convex bodies

Abstract

In this paper, we consider the flag manifold of $p$ orthogonal subspaces of equal dimension which carries an action of the cyclic group of order $p$. We provide a complete calculation of the associated Fadell-Husseini index. This may be thought of as an odd primary version of the computations of Barali\'c et al [Forum Math., 30 (2018), pp. 1539--1572] for the Grassmann manifold $G_n(\mathbb{R}^{2n})$. These results have geometric consequences for $p$-fold orthogonal shadows of a convex body.