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Non-crossing partitions for exceptional hereditary curves

Barbara Baumeister, Igor Burban, Georges Neaime, Charly Schwabe

TL;DR

This work develops reflection groups of canonical type from exceptional hereditary curves and provides a categorical realization of non-crossing partitions via thick subcategories generated by exceptional sequences in Coh(X). It proves Hurwitz transitivity for reduced reflection factorizations, enabling poset isomorphisms between Ex(Coh(X)) and NC_T(W,c) (or its tubular hyperbolic extension), thereby connecting representation-theoretic data with Coxeter-theoretic combinatorics. The approach hinges on canonical bilinear lattices, quotient and hyperbolic extensions, and a robust perpendicular calculus for exceptional sequences, yielding a unified categorification across domestic, tubular, and wild regimes. The results deepen the link between non-commutative geometry, tilting theory, and non-crossing partition theory, with potential implications for elliptic and affine root systems and their Coxeter structures.

Abstract

We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of non-crossing partitions for any such group, realized via the poset of thick subcategories of the category of coherent sheaves on an exceptional hereditary curve generated by an exceptional sequence. A second principal result, essential for the categorification, is a proof of the transitivity of the Hurwitz action in these reflection groups.

Non-crossing partitions for exceptional hereditary curves

TL;DR

This work develops reflection groups of canonical type from exceptional hereditary curves and provides a categorical realization of non-crossing partitions via thick subcategories generated by exceptional sequences in Coh(X). It proves Hurwitz transitivity for reduced reflection factorizations, enabling poset isomorphisms between Ex(Coh(X)) and NC_T(W,c) (or its tubular hyperbolic extension), thereby connecting representation-theoretic data with Coxeter-theoretic combinatorics. The approach hinges on canonical bilinear lattices, quotient and hyperbolic extensions, and a robust perpendicular calculus for exceptional sequences, yielding a unified categorification across domestic, tubular, and wild regimes. The results deepen the link between non-commutative geometry, tilting theory, and non-crossing partition theory, with potential implications for elliptic and affine root systems and their Coxeter structures.

Abstract

We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of non-crossing partitions for any such group, realized via the poset of thick subcategories of the category of coherent sheaves on an exceptional hereditary curve generated by an exceptional sequence. A second principal result, essential for the categorification, is a proof of the transitivity of the Hurwitz action in these reflection groups.

Paper Structure

This paper contains 20 sections, 79 theorems, 186 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Bilinear lattices
  3. Generalities on bilinear lattices and associated reflection groups
  4. Estimates of the reflection length
  5. Exceptional sequences in hereditary and derived categories
  6. Exceptional hereditary curves
  7. Generalities on non-commutative curves
  8. Homogeneous Curves
  9. Combinatorial Parameters
  10. Complete Exceptional Sequences
  11. Reflection groups of canonical type
  12. Introduction to reflection groups of canonical type
  13. Quotient Coxeter group
  14. Hyperbolic Extension
  15. Hurwitz transitivity
  16. ...and 5 more sections

Key Result

Lemma 2.6

For any $\alpha \in \Pi$ we have $s_\alpha^2 = \mathbbm{1}$ and $s_\alpha \in \mathsf{O}(\Gamma, B)$. Moreover, for any other $\beta \in \Pi$ we have

Figures (4)

  • Figure 1: Dynkin diagram for a reflection group of canonical type in the case $\varepsilon=1$.
  • Figure 2: Dynkin diagram for a reflection group of canonical type in the case $\varepsilon=2$.
  • Figure 3: The conventions giving the symmetric bilinear form between two simple roots $\alpha,\beta$ in a canonical lattice.
  • Figure 4: The star-like Dynkin diagram associated with $W_\circ$.

Theorems & Definitions (197)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 187 more