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arXiv · 2105.00237

Equivariant triangulations of tori of compact Lie groups and hyperbolic extension to non-crystallographic Coxeter groups

Arthur Garnier
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Abstract

Given a simple connected compact Lie group $K$ and a maximal torus $T$ of $K$, the Weyl group $W=N_K(T)/T$ 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, using compact hyperbolic extensions rather than affine ones, we construct a compact $W$-manifold $\mathbf{T}(W)$, which is an analogue of a torus for $W$. We exhibit a $W$-equivariant triangulation of $\mathbf{T}(W)$ and compute the associated $W$-dg-ring. Also, we derive its homology representation.

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Arthur Garnier. 2021-05-01. Equivariant triangulations of tori of compact Lie groups and hyperbolic extension to non-crystallographic Coxeter groups. https://doi.org/10.1016/j.jalgebra.2023.07.046

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