A survey of the Shi arrangement

TL;DR

This survey introduces the Shi arrangement, a key hyperplane configuration in algebraic combinatorics, discussing its origins, properties, and significance, especially related to Lusztig's affine permutation map and conjectures.

Contribution

It provides a gentle, accessible overview of the Shi arrangement, summarizing its properties, historical context, and connections to Lusztig's conjecture for readers with modest backgrounds.

Findings

Introduces the Shi arrangement and its properties.

Explains its role in algebraic combinatorics.

Summarizes Lusztig's conjecture and related results.

Abstract

In 1983, Lusztig defined a map $\sigma$ from affine permutations of $n$ to partitions of $n$. He conjectured that for any partition $\lambda$ of $n$, $\sigma^{-1}(\lambda)$ is a two-sided cell. Shi, in 1986, proved part of this conjecture. As a byproduct, he introduced the Shi arrangement of hyperplanes and found a few of its remarkable properties. The Shi arrangement has since become a central object in algebraic combinatorics. This article is intended to be a fairly gentle introduction to the Shi arrangement, intended for readers with a modest background in combinatorics, algebra, and Euclidean geometry.