Vertices of Schubitopes

TL;DR

This paper develops a combinatorial method to generate vertices of Schubitopes, a family of polytopes related to key polynomials, and confirms a conjecture about their structure using Bruhat order permutations.

Contribution

It introduces a new combinatorial rule for generating Schubitope vertices and proves a conjecture linking these vertices to permutations in Bruhat order.

Findings

Vertices of Schubitopes can be generated by permutations in a Bruhat order interval.

Confirmed the conjecture relating key polynomial vertices to Bruhat order permutations.

Provided a combinatorial rule for vertex generation of Schubitopes.

Abstract

Schubitopes were introduced by Monical, Tokcan and Yong as a specific family of generalized permutohedra. It was proven by Fink, M\'esz\'aros and St.$\,$Dizier that Schubitopes are the Newton polytopes of the dual characters of flagged Weyl modules. Important cases of Schubitopes include the Newton polytopes of Schubert polynomials and key polynomials. In this paper, we develop a combinatorial rule to generate the vertices of Schubitopes. As an application, we show that the vertices of the Newton polytope of a key polynomial can be generated by permutations in a lower interval in the Bruhat order, settling a conjecture of Monical, Tokcan and Yong.