Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Barbara Baumeister, Patrick Wegener, Sophiane Yahiatene

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.

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

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 there exists an order preserving bijection between the thick subcategories of generated by an exceptional sequence and a subposet of the interval poset of a Coxeter transformation in the Weyl group of a simply-laced extended root system if the Hurwitz action is transitive on the reduced reflection factorizations of 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 of tubular type.

Paper Structure

This paper contains 34 sections, 62 theorems, 141 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Extended root systems and extended Weyl groups
  3. Some basic notation
  4. Extended Coxeter-Dynkin diagrams
  5. Extended spaces, extended root systems and extended Weyl groups
  6. The Hurwitz action
  7. The reflection length of Coxeter transformations
  8. The interval posets $[\mathop{\mathrm{id}}\nolimits, c]$ and $[\mathop{\mathrm{id}}\nolimits, c]^{\operatorname{gen}}$
  9. The extended root systems attached to the categories $\mathop{\mathrm{coh}}\nolimits(\mathop{\mathrm{\mathbb{X}}}\nolimits)$ and $D^b(\mathop{\mathrm{coh}}\nolimits(\mathbb{X}))$
  10. The categories $\mathop{\mathrm{coh}}\nolimits(\mathbb{X})$ and $D^b(\mathop{\mathrm{coh}}\nolimits(\mathbb{X}))$
  11. Exceptional sequences in $\mathop{\mathrm{coh}}\nolimits(\mathbb{X})$
  12. The extended root system and the extended Weyl group attached to $\mathop{\mathrm{\mathbb{X}}}\nolimits$
  13. The reflection related to an exceptional object
  14. An extended Coxeter-Dynkin diagram for $\mathop{\mathrm{\mathbb{X}}}\nolimits$
  15. An order preserving bijection
  16. ...and 19 more sections

Key Result

Proposition 1.1

Let $(W,S)$ be an extended Weyl system and $c$ a Coxeter transformation in $W$. Then $\ell_T(c)$ equals the rank $\mathrm{rk}(W) = |S|$ of the extended Weyl system.

Figures (6)

  • Figure 1: Extended Coxeter-Dynkin diagram $\Gamma$$\space$
  • Figure 2: One point extended star quiver $\space$
  • Figure 3: (Extended) Dynkin diagrams. Each vertex is labeled by the corresponding coefficient $m_t$ in the linear combination of the highest root in the simple roots.
  • Figure 4: Dynkin diagrams: Numbering $\space$
  • Figure 5: Elliptic Dynkin diagrams for the tubular elliptic root systems
  • ...and 1 more figures

Theorems & Definitions (137)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 127 more