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arXiv · 2012.06841

The hull metric on Coxeter groups

Christian Gaetz·Yibo Gao
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Abstract

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.

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Christian Gaetz, Yibo Gao. 2020-12-12. The hull metric on Coxeter groups. https://doi.org/10.5070/c62257870

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