Generating Functions of Key Polynomials and Bounded Ascending Sequences of Integers

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. In this talk we present a work in progress with Noah Cape investigating 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 can also determine the first terms of the polynomial numerator, and we introduce resulting conjectural relations between the coefficients in key polynomials and signed sums of numbers of integral points on polytopes.