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On certain Coxeter lattices without perfect sections. (English) Zbl 1054.05093

Summary: We compute the kissing numbers of the sections of the Coxeter lattices \(\mathbb A_{n}^{\frac {n+1} 2}\), \(n\) odd, and in particular we prove that for \(n \geqslant 7\) they cannot be perfect. The proof is merely combinatorial and relies on the structure of graphs canonically attached to the sections.

MSC:

05C99 Graph theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05B40 Combinatorial aspects of packing and covering
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