New Bounds for Integer Flows and Verma Modules, via Denormalized Lorentzian Laurent Series

TL;DR

This paper introduces denormalized Lorentzian Laurent series, generalizing Lorentzian polynomials, to derive new bounds for integer flows in graphs and dimensions of Verma modules.

Contribution

It develops the theory of DL Laurent series and applies it to obtain novel bounds in combinatorics and representation theory.

Findings

New bounds for integral flows on directed acyclic graphs

Bounds for dimensions of Verma module weight spaces

Generalization of Lorentzian polynomials to Laurent series

Abstract

The theory of log concave polynomials has recently been developed to study objects and problems in combinatorics and other subfields in mathematics. Particular classes of log concave polynomials called Lorentzian polynomials and denormalized and dually Lorentzian polynomials have been used to prove log concavity statements for various combinatorial sequences. This includes the strongest form of Mason's log concavity conjecture on the independent sets of matroids and the log concavity of sequences of Kostka numbers. In this paper, we develop an analogous class of power series called denormalized Lorentzian (DL) Laurent series. This class is the natural generalization of DL polynomials to homogeneous power series with the benefit of capturing a number of combinatorial generating series including the Kostant partition function for integer flows of directed graphs. We then analyze specific DL Laurent series to obtain new bounds for integral flows on general directed acyclic graphs and new bounds for the dimensions of weight spaces of parabolic $\mathfrak{sl}_{n+1}(\mathbb{C})$ Verma modules.