[Paper Review] Shi variety corresponding to an affine Weyl group

Nathan Chapelier‐Laget|arXiv (Cornell University)|Oct 9, 2020

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper establishes a bijection between an affine Weyl group $W_a$ and the integral points of an affine variety $\widehat{X}_{W_a}$, termed the Shi variety, using Jian-Yi Shi's characterization of alcoves via $\Phi^+$-tuples of integers $k(w,\alpha)$. The key contribution is Theorem 4.3, which shows $W_a \simeq \widehat{X}_{W_a}(\mathbb{Z})$, revealing a geometric realization of the group structure through algebraic geometry and linking it to Kazhdan-Lusztig cells via sign types.

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 $\widehat{X}_{W_a}$, 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 highlight combinatorial properties of the irreducible components of $\widehat{X}_{W_a}$ and we show how they are related to a fundamental parallelepiped $P_{\mathcal{H}}$.

Motivation & Objective

• To establish a geometric realization of affine Weyl groups via an affine variety, termed the Shi variety, to better understand their combinatorial and algebraic structure.
• To extend Jian-Yi Shi's characterization of alcoves in affine Weyl groups to a global algebraic framework using $\Phi^+$-tuples of integers $k(w,\alpha)$.
• To investigate the irreducible components of the Shi variety and their relationship to the fundamental parallelepiped $P_{\mathcal{H}}$ and cell theory.
• To clarify the connection between the Shi variety and Kazhdan-Lusztig cells, particularly in type $A$, by analyzing sign types and generalized orthants.
• To demonstrate that the action of the finite Weyl group $W$ on the Shi variety partitions it into $f_\Phi$ orbits, each containing a unique element of $W$.

Proposed method

• Using Shi's characterization of alcoves in $W_a$ via $\Phi^+$-tuples $(k(w,\alpha))_{\alpha \in \Phi^+}$, the paper defines the Shi variety $\widehat{X}_{W_a}$ as the set of integral points satisfying linear relations derived from the root system.
• The variety $\widehat{X}_{W_a}$ is constructed as a union of affine subspaces, with irreducible components corresponding to generalized orthants in $\mathbb{R}^{\Phi^+}$ defined by sign conditions on the $k(w,\alpha)$.
• The paper employs the $\Phi^+$-representation of $W_a$ to map each group element $w$ to a point $\iota(w) \in \mathbb{Z}^{\Phi^+}$, forming the bijection $W_a \simeq \widehat{X}_{W_a}(\mathbb{Z})$.
• It analyzes the action of the finite Weyl group $W$ on $\widehat{X}_{W_a}$, showing that each orbit contains exactly one element of $W$, and that there are $f_\Phi$ such orbits, where $f_\Phi$ is the number of $W$-orbits on $\Phi^+ \setminus \Delta$.
• The proof of component distinctness for different generators relies on constructing a root $\theta \in \Phi^+ \setminus \Delta$ such that $P_\theta(s_i) \neq P_\theta(s_j)$, leading to $\lambda_\theta(s_i) \neq \lambda_\theta(s_j)$, which contradicts equality of components unless $s_i = s_j$.
• The study of the fundamental parallelepiped $P_{\mathcal{H}}$ provides a geometric model for the irreducible components of $\widehat{X}_{W_a}$, linking them to the geometry of the root system.

Experimental results

Research questions

• RQ1 Can the elements of an affine Weyl group $W_a$ be naturally parameterized by integral points of an algebraic variety?
• RQ2 How are the irreducible components of the Shi variety $\widehat{X}_{W_a}$ related to the fundamental parallelepiped $P_{\mathcal{H}}$ and the root system structure?
• RQ3 What is the role of sign types and generalized orthants in encoding Kazhdan-Lusztig cells within the Shi variety?
• RQ4 Why do distinct simple reflections $s_i$ and $s_j$ lie in different irreducible components of $\widehat{X}_{W_a}$ when $W_a \neq W(\widetilde{A}_2)$?
• RQ5 How does the action of the finite Weyl group $W$ on $\widehat{X}_{W_a}$ decompose the variety into orbits, and what is the significance of the number $f_\Phi$?

Key findings

• The paper establishes a canonical bijection $W_a \simeq \widehat{X}_{W_a}(\mathbb{Z})$, proving that the affine Weyl group is in one-to-one correspondence with the integral points of the Shi variety.
• The irreducible components of $\widehat{X}_{W_a}$ are affine subspaces, each corresponding to a generalized orthant in $\mathbb{R}^{\Phi^+}$ defined by the sign of the $k(w,\alpha)$ coefficients.
• Each Kazhdan-Lusztig cell in type $A$ corresponds to a union of intersections of $\widehat{X}_{W_a}(\mathbb{Z})$ with generalized orthants, as determined by the sign map $Sg(w)$.
• For $W_a \neq W(\widetilde{A}_2)$, distinct simple reflections $s_i$ and $s_j$ lie in different irreducible components of $\widehat{X}_{W_a}$, as shown by constructing a root $\theta$ for which $P_\theta(s_i) \neq P_\theta(s_j)$.
• The action of the finite Weyl group $W$ on $\widehat{X}_{W_a}$ yields exactly $f_\Phi$ orbits, each containing a unique element of $W$, where $f_\Phi$ is the number of $W$-orbits on $\Phi^+ \setminus \Delta$.
• The fundamental parallelepiped $P_{\mathcal{H}}$ provides a geometric model for the irreducible components of $\widehat{X}_{W_a}$, with each component arising as a translate of $P_{\mathcal{H}}$ under the action of $\mathbb{Z}\Phi$.