论文标题
从盒子到多项式：概括的故事
From boxes to polynomials: a story of generalisation
论文作者
论文摘要
在这里，我们将从表面上一些不吉利的盒子开始踏上旅程。小心地以不同的方式堆叠它们会产生惊人的身份。从整数版本的谦虚开始：“从low $ i $到low $ j $需要多少个步骤？”到第一个升级：多项式版本，然后最终到达最终升级：椭圆版本。每个升级都比以前更一般的定理。秘密地，一切都由对称的MacDonald多项式控制。将$ q = t $设置为MacDonald多项式将定理的椭圆形版本带到多项式版本。然后，让$ t $近似$ 1 $将多项式版本降低到整数版本。所有美丽的定理和想法仅来自堆叠框。
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.
