Stabilizers in Coxeter groups with a transpositional action

Noa Cohen; Uzi Vishne

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arXiv · 2609.01309

Stabilizers in Coxeter groups with a transpositional action

Noa Cohen·Uzi Vishne

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Abstract

We study simply-laced Coxeter groups whose Coxeter diagram is the line graph $\mathcal{L}\Gamma$ of some simple graph $\Gamma$; such groups act on the set of vertices of the graph by transpositions. Our goal is to describe the point stabilizer of this action in detail. For any tree $\Sigma$, we show that the stabilizer is a finite-index reflection subgroup, and identify its Coxeter diagram, thereby discovering finite-index embeddings of Coxeter groups $W(\mathcal{L}^{[2]}\Sigma) \hookrightarrow W(\mathcal{L}\Sigma)$ for any tree $\Sigma$, including infinitely many cases where the stabilizer itself is simply-laced.

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Noa Cohen, Uzi Vishne. 2026-09-01. Stabilizers in Coxeter groups with a transpositional action. https://arxiv.org/abs/2609.01309

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