Abstract
We show that the symmetry groups of the cut cone Cut n and the metric cone Met n both consist of the isometries induced by the permutations on \(\{1,\dots,n\}\), that is, \(Is(\mathrm{Cut}{n})=Is(\mathrm{Met}{n})\simeq Sym{n}\) for n ≥ 5. For n = 4 we have \(Is(\mathrm{Cut}{4})=Is(\mathrm{Met}{4})\simeq Sym{3}\times Sym{4}\). This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, \( Is ({\rm Hyp}_n) \simeq Sym(n)\) for n ≥ 5, where Hypn denotes the hypermetric cone.
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Deza, A., Goldengorin, B. & Pasechnik, D.V. The isometries of the cut, metric and hypermetric cones. J Algebr Comb 23, 197–203 (2006). https://doi.org/10.1007/s10801-006-6924-6
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DOI: https://doi.org/10.1007/s10801-006-6924-6