The Frobenius complexity of Hibi rings (Q1789653)

This nice article presents a method to compute the (asymptotic) Frobenius complexity developed by \textit{F. Enescu} and \textit{Y. Yao} [J. Algebra 459, 133--156 (2016; Zbl 1401.13013)] for Hibi rings of characteristic \(p\) which are anticanonical level but not Gorenstein. Given a poset \(P\) on a finite set \(V=\{v_1, \ldots, v_n\}\), denote the set of ideals of \(P\) by \(\mathcal{J}(P)\). The Hibi ring associated to \(P\) is the toric subring \(\mathcal{R}_k[\mathcal{J}(P)]\) of \(k[t,x_{v_1}, \ldots, x_{v_n}]\) generated by the monomials \(t\prod_{v_i \in I} x_{v_i}\) for all \(I\in \mathcal{J}(P)\). Let \(\hat{P}=P \cup \{-\infty, \infty\}\). The main result states that if \(R=\mathcal{R}_{\mathbb{F}_p}[\mathcal{J}(P)]\) is anticanonical level but not Gorenstein, then the asymptotic Frobenius complexity of \(R\) is the cardinality of the set of vertices of \(\hat{P}\) that do not lie on a minimal length upwards path from \(-\infty\) to \(\infty\) which is also one less than the spread of \(\omega^{(-1)}\) where \(\omega\) is the canonical module. Since the Segre product of \(k[x_1,\ldots, x_n]\) with \(k[y_1, \ldots y_m]\) is just a Hibi ring on the poset represented by the disjoint union of an \(n-1\) chain and an \(m-1\) chain which is anticanonical level. If we assume that \(m>n \geq2\) then the vertices which do not lie in a minimal path from \(-\infty\) to \(\infty\) are those lying in the \(m-1\) chain. Hence, the author recovers the result of Enescu and Yao that the Frobenius complexity of a Segre product \(S_{m, n}\) with \(m>n\) is \(m-1\). The paper includes some nice examples and ends with an open question: Is the asympototic Frobenius complexity of a Hibi ring always one less than the spread of \(\omega^{(-1)}\)?