Dirichlet-Voronoi domain and injectivity radius of flag manifolds -- equivariant cell structure on $O(3)/O(1)^3$

TL;DR

This paper develops a method to construct equivariant cell structures on flag manifolds using Dirichlet-Voronoi domains and injectivity radius calculations, exemplified by a new cell structure on a specific flag manifold.

Contribution

It introduces a general approach to build equivariant cell structures on flag manifolds from Dirichlet-Voronoi domains, including explicit computations for particular cases.

Findings

Computed injectivity radius for real and complex flag manifolds.

Established a new $rak{S}_3$-equivariant cell structure on $O(3)/O(1)^3$.

Derived cellular complex of $bZ[rak{S}_3]$-modules for the flag manifold.

Abstract

In the first part of this work, we study Dirichlet-Voronoi domains for discrete isometry groups of Riemannian manifolds, in view of constructing cell structures on homogeneous (complete, real) flag manifolds, equivariant with respect to the action of the Weyl group. We give general results, allowing to build such a structure from an admissible one on the domain. In particular, the injectivity radius plays a key role in the method. The second part starts with the computation of the injectivity radius of (real and complex) flag manifolds; a first step towards the application of the method developed in the first part. Then, with the help of the quaternion algebra, we investigate the particular case of the flag manifold $O(3)/O(1)^3$ of $SL_3(\mathbb{R})$: we prove that the results of the first part apply and derive a new $\mathfrak{S}_3$-equivariant cell structure on it, whose cellular complex of $\mathbb{Z}[\mathfrak{S}_3]$-modules is determined.