
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 from affine permutations of to partitions of . He conjectured that for any partition of , 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Analytic Number Theory Research
