Let \((h_ 0, \dots, h_ d)\) be the \(h\)-vector of a simplicial \(d\)- polytope \(P\). A result by Stanley states that if \(P\) is centrally symmetric, then for all \(i\), \(1 \leq i \leq d/2\), \[ h_ i - h_{i-1} \geq {d \choose i} - {d \choose i - 1}. \] The author extends this result to simplicial rational polytopes admitting a fixed-point-free linear group action of a cyclic group \(G\) of prime-power order \(n = p^ \nu\). Defining \(P_{\min} (G,d)\) as the free sum of \(d/(p - 1)\) copies of the \((p - 1)\)-simplex and \[ \sum^ d_{i=0} a_ iq^ i = h_ P(q) - h_{P_{\min} (G,D)} (q) \] as the difference between the corresponding \(h\)-polynomials, the main result says that the coefficients \(a_ 0, \dots, a_ d\) are symmetric, divisible by \(n\), nonnegative and unimodal.
The main part of the proof consists in establishing the unimodality of the coefficients of the polynomial \(h_ P (q) - h_{P_{\min} (G,d)} (q)\). To this effect this polynomial is realized as the Hilbert- Poincaré series of the isotypical component, corresponding to a suitable irreducible character of \(G\), of the cohomology ring \(H^* (X_ P; \mathbb{C})\) of the toric variety \(X_ P\) associated to the polytope \(P\). All definitions and results needed are recalled in an introductory section.
A possible extension to cyclic groups of non prime power order is also considered in a final section.
For the entire collection see [Zbl 0806.00023].