[Paper Review] Exceptional sequences and clusters

Kiyoshi Igusa, Ralf Schiffler|arXiv (Cornell University)|Jan 16, 2009

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper establishes a deep connection between exceptional sequences in hereditary algebras and cluster-tilting sets in cluster categories by showing that a sequence of reflections corresponds to a complete exceptional sequence if and only if their product equals the inverse Coxeter element. The key contribution is a new combinatorial criterion: a set of $ n $ objects in the cluster category forms a cluster-tilting set if and only if the deleted word from a reduced expression of $ w_0 $ is itself reduced, providing a type-independent characterization of cluster-tilting objects via reflection group combinatorics.

ABSTRACT

We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type.

Motivation & Objective

• To characterize complete exceptional sequences in hereditary algebras using the product of corresponding reflections.
• To establish a combinatorial criterion for cluster-tilting sets in the cluster category of a finite-type hereditary algebra.
• To unify and generalize prior results on exceptional sequences and cluster tilting in finite and affine types.
• To prove that a real root is a real Schur root if and only if its corresponding reflection is a prefix of the Coxeter element.
• To show that the map from extension-closed subcategories of $ \mathrm{mod}\,kQ $ to prefixes of the Coxeter element is a bijection.

Proposed method

• Use the braid group action on sequences of reflections whose product is the inverse Coxeter element $ C^{-1} $, showing transitivity via Theorem 1.4.
• Construct a reduced expression for the longest element $ w_0 $ in the Weyl group using reflections from the Auslander-Reiten quiver of a hereditary algebra.
• Define a deleted word $ w^\delta(t_1,\dots,t_n) $ by removing $ n $ reflections from the expression $ s_1\cdots s_n s_{i_1}\cdots s_{i_\nu} $, which represents $ C w_0 $.
• Prove that $ w^\delta $ is a reduced expression for $ w_0 $ if and only if the corresponding set of $ n $ objects in the cluster category forms a cluster-tilting set (Theorem 2.5).
• Use algebraic mutation of exceptional sequences to model cluster mutation, linking the two concepts combinatorially.
• Apply the main result to show that a real root is a real Schur root iff its reflection is a prefix of the Coxeter element (Corollary 4.2).

Experimental results

Research questions

• RQ1 When does a sequence of reflections in the Weyl group correspond to a complete exceptional sequence in a hereditary algebra?
• RQ2 What combinatorial condition on a word derived from a Coxeter element ensures that the corresponding set of objects in the cluster category is cluster-tilting?
• RQ3 How can the braid group action be used to characterize exceptional sequences and cluster-tilting objects uniformly across types?
• RQ4 Under what conditions is a real root a real Schur root, and how does this relate to the Coxeter element's factorization?
• RQ5 Is there a bijection between extension-closed subcategories of $ \mathrm{mod}\,kQ $ and prefixes of the Coxeter element?

Key findings

• A sequence of $ n $ reflections whose product is $ C^{-1} $ corresponds to a complete exceptional sequence if and only if the braid group acts transitively on such sequences, as established in Theorem 1.4.
• A set of $ n $ indecomposable objects in the cluster category of a finite-type hereditary algebra forms a cluster-tilting set if and only if the corresponding deleted word from the expression $ s_1\cdots s_n s_{i_1}\cdots s_{i_\nu} $ is a reduced expression for $ w_0 $, as shown in Theorem 2.5.
• The mutation of cluster-tilting sets corresponds to specific braid group moves on the reduced expression $ w^\delta(t_1,\dots,t_n) $, as formalized in Theorem 2.8.
• A real root $ \beta $ is a real Schur root if and only if the reflection $ s_\beta $ is a prefix of the Coxeter element $ C $, as proven in Corollary 4.2.
• The map $ \phi $ from finitely generated, exact abelian, extension-closed subcategories of $ \mathrm{mod}\,kQ $ to prefixes of the Coxeter element is a bijection, as established in Theorem 4.3.
• The deleted word construction generalizes to the infinite case and provides a uniform framework for characterizing exceptional sequences and cluster-tilting objects across all types.