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Para-exceptional sequences for tame hereditary algebras and McCammond-Sulway lattices

Eric J. Hanson, Nathan Reading

Abstract

Noncrossing partition posets in a Coxeter group $W$ can fail to be lattices when $W$ is not finite. When the lattice property fails for $W$ of affine type, McCammond and Sulway's construction provides a larger lattice that contains the noncrossing partition poset and that furthermore is a combinatorial Garside structure. We construct a lattice, isomorphic to McCammond and Sulway's lattice, using the representation theory of a corresponding connected tame hereditary algebra and give a representation-theoretic proof that it is a combinatorial Garside structure. To construct the lattice, we introduce para-exceptional sequences and para-exceptional subcategories in the module categories of tame hereditary algebras. Para-exceptional sequences are generalizations of exceptional sequences obtained by enlarging the set of allowed entries to include all non-homogeneous bricks. A para-exceptional subcategory is a subcategory obtained by applying a certain closure-like operator to the wide subcategory generated by a para-exceptional sequence.

Para-exceptional sequences for tame hereditary algebras and McCammond-Sulway lattices

Abstract

Noncrossing partition posets in a Coxeter group can fail to be lattices when is not finite. When the lattice property fails for of affine type, McCammond and Sulway's construction provides a larger lattice that contains the noncrossing partition poset and that furthermore is a combinatorial Garside structure. We construct a lattice, isomorphic to McCammond and Sulway's lattice, using the representation theory of a corresponding connected tame hereditary algebra and give a representation-theoretic proof that it is a combinatorial Garside structure. To construct the lattice, we introduce para-exceptional sequences and para-exceptional subcategories in the module categories of tame hereditary algebras. Para-exceptional sequences are generalizations of exceptional sequences obtained by enlarging the set of allowed entries to include all non-homogeneous bricks. A para-exceptional subcategory is a subcategory obtained by applying a certain closure-like operator to the wide subcategory generated by a para-exceptional sequence.
Paper Structure (23 sections, 105 theorems, 47 equations, 2 figures)

This paper contains 23 sections, 105 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization and main results
  3. Acknowledgments
  4. Background
  5. The noncrossing partition poset in a Coxeter group
  6. Hereditary algebras
  7. Exceptional sequences
  8. Wide subcategories
  9. Chain systems
  10. Definition of a chain system
  11. Chain systems and labeled posets
  12. The shuffle product of chain systems
  13. Chain systems and combinatorial Garside structures
  14. The chain system of exceptional sequences
  15. The chain system of the noncrossing partition poset
  16. ...and 8 more sections

Key Result

Theorem 2.2

The map $(X_1,\ldots,X_n) \mapsto t_{\underline{\dim} X_1}\cdots t_{\underline{\dim} X_n}$ is a bijection from the set of complete exceptional sequences of $\operatorname{mod}\Lambda$ to the set of reduced $T$-words for $c$.

Figures (2)

  • Figure 1: Some noncrossing partitions of an annulus
  • Figure 2: Illustrations of the proof of Proposition \ref{['good bij']}

Theorems & Definitions (187)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 177 more