Shi variety corresponding to an affine Weyl group

2020/10/09 by Nathan Chapelier‐Laget, Chapelier-Laget, Nathan · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2010.04310

openalex publication_date 2020/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let W be an irreducible Weyl group and W_a its affine Weyl group. In this article we show that there exists a bijection between W_a and the integral points of an affine variety, denoted \widehatX_Wa, which we call the Shi variety of W_a. In order to do so, we use Jian-Yi Shi's characterization of alcoves in affine Weyl groups. We then study this variety further. We highlight combinatorial properties of the irreducible components of \widehatX_Wa and we show how they are related to a fundamental parallelepiped P_H.

Cited by

• Atomic length on Weyl groups

Related

• Lattice associated to a Shi variety
• Orientation of alcoves in affine Weyl groups
• A symmetric group action on the irreducible components of the Shi variety associated to W(\widetildeA_n)
• The Shi arrangement and the Ish arrangement
• A basis construction for the Shi arrangement of the type B_ℓ or C_ℓ