TITLE: Log-concavity of matroids: characterization of equality?
QUESTION [14 upvotes]: Let $M$ be a (loopless) matroid of rank $r$.
The characteristic polynomial $\chi_M(x)$ is defined by $\chi_M(x)=\sum_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(M)$ is the lattice of flats of $M$ and $\mu$ its Möbius function. It is known that the signs of the characteristic polynomial alternate, and the so-called Whitney numbers of the 1st kind $\omega_i$ are defined by $\chi_M(x) = \sum_{i=0}^{r} (-1)^i \omega_i x^i$.
In Hodge theory for combinatorial geometries, affirming a long-standing conjecture of Rota and others, Adiprasito-Huh-Katz (AHK) showed that these $\omega_i$ form a log-concave sequence, i.e., $\omega_i^2 \geq \omega_{i-1}\omega_{i+1}$.
Actually, they proved something stronger. It is known that $(1-x)$ is always a factor of $\chi_M(x)$, and the reduced Whitney numbers $\overline{\omega}_i$ are thus defined by $\chi_M(x)/(1-x)=\sum_{i=0}^{r-1} (-1)^i \overline{\omega}_i x^i$. AHK showed that in fact the reduced Whitney numbers form a log-concave sequence: $\overline{\omega}_i^2 \geq \overline{\omega}_{i-1}\overline{\omega}_{i+1}$ (which implies log-concavity of the $\omega_i$ by a straightforward argument).
In fact, an easy reduction using the truncation of the matroid $M$ shows that it is enough to prove log-concavity of the $\overline{\omega}_i$ in the "last-spot": i.e., that $\overline{\omega}_{r-2}^2 \geq \overline{\omega}_{r-3}\cdot\overline{\omega}_{r-1}$.
To prove this, AHK use the Chow ring $A(M) = A^0(M)\oplus A^1(M)\oplus \cdots\oplus A^{r-1}(M)$ of the matroid $M$: they show that $\overline{\omega}_k=\langle \alpha^{r-1-k}\beta^{k}\rangle$ for certain linear elements $\alpha,\beta\in A^1(M)$ of the Chow ring, where $\langle \cdot \rangle\colon A^{r-1}(M)\to \mathbb{R}$ is the canonical degree map isomorphism; and they deduce $\overline{\omega}_{r-2}^2 \geq \overline{\omega}_{r-3}\cdot\overline{\omega}_{r-1}$ from the so-called Kähler package for $A(M)$, in particular, the Hodge-Riemann relations, which imply that the relevant $2\times 2$ determinant is nonpositive.
Question: From the work of AHK (or elsewhere) is it possible to deduce when (i.e. for which matroids) we have an equality $\overline{\omega}_{r-2}^2 = \overline{\omega}_{r-3}\cdot\overline{\omega}_{r-1}$? (In other words, when the determinant of the $2\times 2$ HR relations matrix is $0$ rather than negative?)

REPLY [6 votes]: I have nothing to add to Hunter's answer to your main question, but I thought it might be helpful to comment more generally on where the difficulty lies in extracting such information from a Kähler package.
Let us adopt the general setting in this review of Huh. Let $A(X)=\bigoplus_{q=0}^d A^q(X)$ be a graded algebra with Kähler cone $\mathrm{K}(X)\subset A^1(X)$, satisfying Poincare duality, hard Lefschetz (HL), and Hodge-Riemann (HR). A simple consequence of (HR) is that for any $\mathrm{L}_0,\ldots,\mathrm{L}_{d-2}\in\mathrm{K}(X)$ and $\eta\in A^1(X)$, we have $$(\eta \mathrm{L}_0\cdots \mathrm{L}_{d-2})^2\ge (\eta^2\mathrm{L}_1\cdots \mathrm{L}_{d-2})\, (\mathrm{L}_0^2\mathrm{L}_1\cdots \mathrm{L}_{d-2}).$$This is what gives rise to log-concavity in various applications. Moreover, (HL) implies that in this inequality, equality holds if and only if $\eta$ is proportional to $\mathrm{L}_0$. There are various ways to see this. For example, if one has equality in the above inequality, then the quadratic polynomial $$q(t) = ((\eta+t\eta') \mathrm{L}_0\cdots \mathrm{L}_{d-2})^2-((\eta+t\eta')^2\mathrm{L}_1\cdots \mathrm{L}_{d-2})\, (\mathrm{L}_0^2\mathrm{L}_1\cdots \mathrm{L}_{d-2})$$satisfies $q(0)=0$ and $q(t)\ge 0$ for all $t$ and $\eta'\in A^1(X)$. Computing $q'(0)=0$ shows that a linear combination of $\eta$ and $\mathrm{L}_0$ lies in the kernel of the map $\mathrm{L}_1\cdots \mathrm{L}_{d-2}:A^1(X)\to A^{d-1}(X)$. But (HL) states this map is a bijection, so its kernel is trivial, and thus $\eta$ and $\mathrm{L}_0$ are proportional.
The difficulty is that in many applications, the $\mathrm{L}_i$ of interest do not lie in $\mathrm{K}(X)$, but can only be approximated by elements of $\mathrm{K}(X)$. This is the case in AHK (see, e.g., the last section of this Notices paper), in the Khovanskii-Teissier inequality when one deals with nef classes rather than ample classes, etc. In such cases, the inequality still follows by approximation, but all information on the equality cases is lost. While no nontrivial equality cases turn out to appear in the particular case of AHK (as per Hunter's answer), many new equality cases can appear for the limiting inequalities in other situations.
As far as I know, the only structures of this kind where the equality cases of the limiting inequalities are fully understood (in the sense that there is a complete geometric/combinatorial characterization) are mixed discriminants and mixed volumes of convex polytopes. The last section of the latter paper contains some speculative remarks on what such results might look like more generally.