The Shi variety corresponding to an affine Weyl group

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-02-13 DOI:10.1112/blms.70007

Nathan Chapelier-Laget

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引用次数: 0

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 ${\hat{X}}_{W_{a}}$ , 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 introduce a new representation of $W_{a}$ , called the $Φ^{+}$ -representation, and we highlight combinatorial and geometrical properties of the irreducible components of ${\hat{X}}_{W_{a}}$ via this representation. We also show how the components are related to a fundamental parallelepiped $P_{H}$ .