Total positivity and two inequalities by Athanasiadis and Tzanaki

Let Δ be a (d-1)-dimensional simplicial complex and h^Δ = (h_0^ ,.. , h_d) its h-vector. For a face uniform subdivision operation we write Δ_ for the subdivided complex and H_ for the matrix such that hΔ_ = H_ h^Δ. In connection with the real rootedness of symmetric decompositions Athanasiadis and Tzanaki studied for strictly positive h-vectors the inequalities h_0 / h_1 ≤ h_1 / hd-1 ≤ .... ≤ h_d / h_0 and h_1 / hd-1 ≥ ... ≥ hd-2 / h_2 ≥ hd-1 / h_1. In this paper we show that if the inequalities holds for a simplicial complex Δ and H_ is TP_2 (all entries and two minors are non-negative) then the inequalities hold for Δ_. We prove that if is the barycentric subdivision then H_ is TP_2. If is the rth-edgewise subdivision then work of Diaconis and Fulman shows H_ is TP_2. Indeed in this case by work of Mao and Wang H_ is even TP.