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The hull metric on Coxeter groups

Published Web Location Commons 'BY' version 4.0 license

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.

Mathematics Subject Classifications: 05A20, 05C12, 05E16, 20F55

Keywords: Linear extension, promotion, Coxeter group, convex hull, metric

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