Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions
Apoorva Khare, Jacob P. Matherne, Avery St. Dizier
TL;DR
The paper proves that shifted, normalized characters of parabolic Verma modules over $\mathfrak{sl}_{n+1}$ (and their semisimple products) are Lorentzian, yielding both continuous and discrete log-concavity in type $A$ root directions. It connects parabolic Vermas to restricted Kostant partition functions via flow polytopes and uses Alexandrov–Fenchel inequalities to deduce discrete log-concavity, with a parallel flow-polytope proof. The results show that parabolic Vermas form a maximal class with log-concave characters, while higher-order Verma modules and non-$A$ types fail to preserve these properties, and they extend HMMS by addressing integral and nonintegral weights and direct-sum algebras. The work also provides counterexamples in Jack/Macdonald settings and outside type $A$, clarifying the limits of log-concavity in representation-theoretic and symmetric-function contexts. Overall, the Lorentzian/flow-polytope framework unifies discrete and continuous log-concavity phenomena for a broad family of highest-weight modules and their associated partition functions, with significant implications for Kostant partition functions and Schur-type polynomials.
Abstract
In 2022, Huh-Matherne-Mészáros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type $A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian (DL). In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are DL. In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing. We then show that parabolic Vermas form a "maximal" class with log-concave (hence DL) characters. Namely, log-concavity fails in greater generality along three natural directions: (1) it does not hold for every simple Lie type, (2) nor for a larger universal family of highest weight modules, the higher order Verma modules, even in type $A$, and (3) it does not always hold for important generalizations of Schur polynomials: the Jack and Macdonald polynomials. Finally, we extend these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$. We also partially resolve a conjecture of Huh et al on the DL property for integral highest weight simple modules.