Polynomials defined by tableaux and linear recurrences

TL;DR

This paper demonstrates that various families of polynomials, including key polynomials and Hall-Littlewood polynomials, satisfy linear recurrences related to diagram shape operations, revealing connections to polytopes and divided difference operators.

Contribution

It introduces a unified approach to establish linear recurrences for multiple polynomial families defined via tableaux, expanding understanding of their structural properties.

Findings

Key polynomials and Demazure atoms satisfy linear recurrences.

Hall-Littlewood and dual Grothendieck polynomials also follow these recurrences.

Recurrences relate to lattice point interpretations and divided difference operators.

Abstract

We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure atoms. The same technique can be applied to Hall-Littlewood polynomials and dual Grothendieck polynomials. The motivation behind this is that such recurrences are strongly connected with other nice properties, such as interpretations in terms of lattice points in polytopes and divided difference operators.