Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case

TL;DR

This paper explores the structure of extended Weyl groups and their relation to hereditary categories, establishing a connection between thick subcategories and Coxeter transformations for weighted projective lines, especially in the tubular case.

Contribution

It introduces a new combinatorial and group-theoretical framework linking thick subcategories to Weyl group elements via Hurwitz transitivity, specifically for tubular weighted projective lines.

Findings

Established a bijection between thick subcategories and subposets of interval posets of Coxeter transformations.

Proved Hurwitz action transitivity for tubular weighted projective lines.

Extended the understanding of Weyl groups in the context of hereditary categories.

Abstract

We start the systematic study of extended Weyl groups, and continue the combinatorial description of thick subcategories in hereditary categories started by Ingalls-Thomas, Igusa-Schiffler-Thomas and Krause. We show that for a weighted projective line $\mathbb{X}$ there exists an order preserving bijection between the thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and a subposet of the interval poset of a Coxeter transformation $c$ in the Weyl group of a simply-laced extended root system if the Hurwitz action is transitive on the reduced reflection factorizations of $c$ that generate the Weyl group. By using combinatorial and group theoretical tools we show that this assumption on the transitivity of the Hurwitz action is fulfilled for a weighted projective line $\mathbb{X}$ of tubular type.