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 us 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 SL3(R): we prove that the results of the first part apply and derive a new C53-equivariant cell structure on it, whose cellular complex of Z[C53]-modules is determined.

A fundamental alcove d is a tile in a paving of a vector space V by an affine reflection group Waff. Its geometry encodes essential features of Waff, such as its affine Dynkin diagram Dr and fundamental group Omega. In this article we investigate its full isometry group Aut(d). It is well known that the isometry group of a regular polyhedron is generated by hyperplane reflections on its faces. Being a simplex, an alcove d is the simplest of polyhedra, nevertheless it is seldom a regular one. In our first main result we show that Aut(d) is isomorphic to Aut(Dr). Building on this connection, we establish that Aut(d) is an abstract Coxeter group, with generators given by affine isometric involutions of the ambient space. Although these involutions are seldom reflections, our second main result leverages them to construct, by slicing the Komrakov-Premet fundamental polytope X' for the action of Omega, a family of fundamental polytopes for the action of Aut(d) on d whose vertices are contained in the vertices of X' and whose faces are parametrized by the so-called balanced minuscule roots, which we introduce here. In an appendix, we discuss some related negative results on stratified centralizers and equivariant triangulations. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

AbstractIn this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity $$v>1$$ v > 1 and we give approximate values for their size when $$v > rsim 1.$$ v ≳ 1 . For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.

We provide a fundamental domain for the action of the finite Weyl group on a maximal torus of a compact Lie group of the corresponding type. The general situation is reduced to the adjoint case and, from the perspective of root data, this problem can be rephrased by asking for a fundamental polytope for the action of the extended affine Weyl group on the (dual) toral subalgebra. We solve the problem in this second form. Using the theory of minuscule weights, we obtain a description of this fundamental polytope as a convex hull of explicit vertices, and as an intersection of closed half-spaces. The latter description was first obtained by Komrakov and Premet in 1984 but, as the present work is independent of that of Komrakov-Premet, we give a new self-contained proof of it. We also derive some consequences on the structure of automorphism groups of extended Dynkin diagrams.

Given a simple connected compact Lie group K and a maximal torus T of K, the Weyl group W = NK(T)/T naturally acts on T. First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit W-equivariant triangulation of T. We describe the associated W -dg-ring. For a non-crystallographic Coxeter group W, using compact hyperbolic extensions rather than affine ones, we construct a compact W -manifold T(W), which is an analogue of a torus for W. We exhibit a W-equivariant triangulation of T(W) and compute the associated W -dg-ring. Also, we derive its homology representation.& COPY; 2023 Elsevier Inc. All rights reserved.

In this paper, we recall some basic facts about the Kerr–Newman–(anti) de Sitter (KNdS) spacetime and review several formulations and integration methods for the geodesic equation of a test particle in such a spacetime. In particular, we introduce some basic general symplectic integrators in the Hamiltonian formalism and we re-derive the separated motion equations using Carter’s method. After this theoretical background, we explain how to ray-trace a KNdS black hole, equipped with a thin accretion disk, using Scilab. We compare the accuracy and execution time of the previous methods, concluding that the Carter equations is the best one. Then, inspired by Hagihara, we apply Weierstrass’ elliptic functions to the non-rotating case, yielding a fairly fast shadowing program for such a spacetime. We provide some illustrations of the code, including a depiction of the effects of the cosmological constant on shadows and accretion disk, as well as a simulation of M87*.

We construct an explicit equivariant cellular decomposition of the $(4n-1)$-sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex. As a corollary of the binary octahedral case, we deduce an $\mathfrak{S}_3$-equivariant decomposition of the flag manifold of $SL_3(\mathbb{R})$.

We present some perspectives in the construction of explicit cell structures on real flag manifolds, equivariant with respect to the (free) action of the Weyl group. Such structures could be obtained from Dirichlet-Voronoi fundamental domains associated to these manifolds, defined using normal homogeneous metrics. First, we motivate the study by reviewing the Riemannian geometric properties of the flag manifold $\mathcal{F}_3(\mathbb{R})=O(3)/O(1)^3$ of $SL_3(\mathbb{R})$ and exhibit some geodesic properties of an $\mathfrak{S}_3$-equivariant cell structure of $\mathcal{F}_3(\mathbb{R})$ previously constructed by R. Chiriv\`i, M. Spreafico and the author. In particular, the 1-cells are seen to be open geodesic arcs. Then, we define Dirichlet-Voronoi domains for Riemannian manifolds, equipped with a finite group of isometries and give their first properties. Under a rather strong condition on the injectivity radius of the manifold, such domains are a reasonable starting point towards the construction of cell structures. We prove moreover that a nice enough cell structure on such a domain induces an equivariant cell structure on the whole manifold. We apply these considerations to produce a new $\mathfrak{S}_3$-equivariant cell structure on $\mathcal{F}_3(\mathbb{R})$.