The Schur Cone and the Cone of Log Concavity

TL;DR

This paper investigates the structure of the cone generated by specific algebraic expressions related to log concavity and Schur functions, proposing a conjecture and partial characterizations of its extreme vectors.

Contribution

It introduces a conjecture on the extreme vectors of the cone of log concavity and provides partial proofs and characterizations for these vectors.

Findings

Proposed a conjecture for extreme vectors in the cone of log concavity.

Proved one direction of the conjecture.

Provided partial results supporting the conjecture.

Abstract

Let $\{h_1,h_2,...\}$ be a set of algebraically independent variables. We ask which vectors are extreme in the cone generated by $h_ih_j-h_{i+1}h_{j-1}$ ($i\geq j>0$) and $h_i$ ($i>0$). We call this cone the cone of log concavity. More generally, we ask which vectors are extreme in the cone generated by Schur functions of partitions with $k$ or fewer parts. We give a conjecture for which vectors are extreme in the cone of log concavity. We prove the characterization in one direction and give partial results in the other direction.