Extreme rays of the $(N, k)$-Schur Cone

TL;DR

This paper investigates the structure of the $(N,2)$-Schur cone, providing new proofs and extending the understanding of its extreme rays, and reduces White's conjecture to simpler related conjectures.

Contribution

It offers an alternative proof for a special case of White's conjecture and identifies additional infinite families of extreme rays, advancing the theoretical understanding of the cone.

Findings

Extended the set of known extreme rays in the $(N,2)$-Schur cone

Provided an alternative proof for a special case of White's conjecture

Reduced White's conjecture to two simpler conjectures

Abstract

We discuss several partial results towards proving Dennis White's conjecture on the extreme rays of the $(N,2)$-Schur cone. We are interested in which vectors are extreme in the cone generated by all products of Schur functions of partitions with $k$ or fewer parts. For the case where $k =2$, White conjectured that the extreme rays are obtained by excluding a certain family of "bad pairs," and proved a special case of the conjecture using Farkas' Lemma. We present an alternate proof of the special case, in addition to showing more infinite families of extreme rays and reducing White's conjecture to two simpler conjectures.