From boxes to polynomials: a story of generalisation

Abstract

Here we will embark on a journey starting with some ostensibly inauspicious boxes. Carefully stacking them in different ways yields amazing identities. From humble beginnings at the integer version: `how many steps does it take to get from row $i$ to row $j$?' to the first upgrade: the polynomial version, before finally reaching the final upgrade: the elliptic version. Each upgrade gives a more general theorem than before. Secretly, everything is controlled by the symmetric Macdonald polynomials. Setting $q = t$ in the Macdonald polynomial takes the elliptic version of the theorem to the polynomial version. Then, letting $t$ approach $1$ reduces the polynomial version to the integer version. All the beautiful theorems and ideas come merely from stacking boxes.