On face numbers of rational simplicial polytopes with symmetry (Q1908488)

The paper contains results on simplicial \(d\)-polytopes \(P\) with a fixed-point-free symmetry and vertices having only rational coordinates with respect to the standard basis of \(R^d\). (Such a symmetry is an invertible linear transformation of \(R^d\) which maps \(P\) onto itself without any fixed point except the origin \(0 \in \text{int } P\).) For an integer \(p > 1\), a \(p\)-symmetry of \(P\) is a fixed-point-free linear symmetry of \(P\) which is of order \(p\) as an element of the symmetry group. (E.g., a polytope is 2-symmetric in this sense if it has usual central symmetry.) Extending a result of R. Stanley on centrally symmetric polytopes, the author obtains tight lower bounds for the \(h\)-vector (and, consequently, for all the face numbers) of rational simplicial \(d\)-polytopes having fixed-point-free linear \(p\)-symmetry for some prime \(p\).