Chiral polytopes of full rank exist only in ranks \(4\) and \(5\) (Q2042130)

Let \({\mathcal P}\) be a skeletal polytope (the definition is recalled in the paper) of rank \(n\) in \({\mathbb E}^d\), and let \(G(\mathcal P)\) be its symmetry group, that is, the group of isometries of \({\mathbb E}^d\) that permute the set of \(i\)-faces of \(\mathcal P\), for each \(i\). Then \(\mathcal P\) is called chiral if \(G(\mathcal P)\) induces two orbits on flags such that adjacent flags belong to distinct orbits (intuitively, a chiral polytope has maximal rotational symmetry, but no reflectional symmetry). The polytope \(\mathcal P\) is said to be of full rank if \(n=d\) if \(\mathcal P\) is finite, and \(n-1=d\) if \(\mathcal P\) is infinite. It is known that chiral polytopes of full rank \(n\) do not exist if \(n=2\) or \(n=3\) and that they do exist for \(n=4\) and \(n=5\). The present paper proves that a chiral polytope of full rank can only have rank \(4\) or \(5\). The proof is achieved after collecting sufficiently many constraints for chiral polytopes. The author mentions some questions on complete classifications that remain open.