Generating Series of Key Polynomials and Bounded Ascending Sequences of Integers

TL;DR

This paper explores the generating functions of key polynomials, revealing their structure involves bounded ascending sequences and conjecturing relations to polytope lattice points, extending understanding beyond Schubert polynomials.

Contribution

It introduces a new analysis of key polynomial generating functions, linking their coefficients to bounded sequences and integral points on polytopes, with conjectures on their general form.

Findings

Explicit first terms of numerator polynomials identified

Denominator structure based on bounded ascending sequences

Conjectures proposed relating coefficients to polytope lattice points

Abstract

The fact that Schubert polynomials are the weighted counting functions for reduced RC-graphs, also known as reduced pipe dreams, was established using their generating functions inside an appropriate Demazure algebra. Here we investigate the generating functions of another family of polynomials, the key polynomials, also known as Demazure characters. Each component in that function is a rational function, whose denominator is an explicit product whose definition is based on bounded ascending sequences of integers. We determine the first terms of the polynomial numerator, and pose conjectures about these terms in general as well as some of the next ones. The form of our generating functions suggests relations between the coefficients in key polynomials and signed sums of numbers of integral points on polytopes.