On the cube problem of Las Vergnas (Q1815252)

Associated with the \(d\)-dimensional (unit) cube one has its matroid \(M^d\) of affine dependencies (of rank \(d+1\), on \(2^d\) elements). As proved here, the symmetry group of \(M^d\) is the Coxeter group \(BC_{d+1}\) given by the symmetries of the \((d+1)\)-cube, modulo its 2-element center. This symmetry information is used to prove for \(d\leq 7\) a conjecture of Las Vergnas: the reorientation class of \(M_d\) is unique. The second component used for this is that the labelled contractions of rank 3 determine every oriented matroid of rank \(r\geq 3\) uniquely; thus it suffices to show that all rank 3 contractions of \(M_d\) have unique reorientation class. For \(d=7\) this is established using computer support, where the knowledge of the symmetry group is used to reduce the number of cases.