Homotopy Types Of Toric Orbifolds From Weyl Polytopes

TL;DR

This paper explores the homotopy types of toric orbifolds derived from Weyl polytopes associated with crystallographic root systems, revealing their topological equivalences under certain lattice considerations.

Contribution

It establishes the homotopy equivalence between quotient spaces of toric varieties from Weyl polytopes and their associated polytopes, providing new insights into their topological structure.

Findings

Homotopy equivalence between $X_P/W_K$ and $X_{P/W_K}$ spaces.

Topological spaces are homotopy equivalent when considering polytopes in root or weight lattice spans.

Clarifies the topological relationship between toric orbifolds and Weyl polytopes.

Abstract

Given a reduced crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$, parabolic subgroups $W_K$'s and a polytope $P$ which is the convex hull of a dominant weight. The quotient $P/W_K$ can be identified with a polytope. Polytopes $P$ and $P/W_K$ are associated to toric varieties $X_P$ and $X_{P/W_K}$ respectively. It turns out the underlying topological spaces $X_P/W_K$ and $X_{P/W_K}$ are homotopy equivalent, when considering the polytopes in the real span of the root lattice or of the weight lattice.