[Paper Review] Exceptional Sequences over path algebras of type $A_n$ and Non-crossing Spanning Trees

Tokuji Araya|arXiv (Cornell University)|Apr 18, 2009

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper establishes a bijection between isomorphism classes of complete exceptional sequences in the bounded derived category of the path algebra of type $A_n$ and non-crossing spanning trees on $n+1$ labeled points on a circle. Using the Auslander-Reiten quiver and a geometric realization of indecomposable objects as chords, the authors show that mutation of exceptional sequences corresponds to a combinatorial operation on these trees, yielding a complete combinatorial classification of exceptional sequences via Catalan-type enumeration.

ABSTRACT

Exceptional sequences are fundamental to investigate the derived categories of finite dimensional algebras. The aim of this note is to classify all the complete exceptional sequences over the path algebra of a Dynkin quiver of type $A_n$ in terms of non-crossing spanning trees.

Motivation & Objective

• To provide a complete combinatorial classification of complete exceptional sequences in the derived category of the path algebra of type $A_n$.
• To establish a bijection between such sequences (up to shift and permutation) and non-crossing spanning trees on $n+1$ points.
• To interpret the mutation of exceptional sequences as a geometric operation on chords and trees.
• To give a concrete, explicit correspondence using the Auslander-Reiten quiver and chord diagrams.

Proposed method

• Representing indecomposable modules in the derived category as chords $c(i,j)$ on a circle with $n+1$ labeled points.
• Defining a map $\Phi$ from the set of indecomposable objects to the set of chords, preserving shifts.
• Using the Auslander-Reiten quiver of $A_n$ to encode the structure of modules and their extensions.
• Defining left and right mutations $\mathcal{L}_E F$ and $\mathcal{R}_F E$ via triangle completions in the derived category.
• Introducing a combinatorial mutation operation on chords: $\mathcal{L}_{c} c' = c(j,l)$ when $c = c(i,j)$, $c' = c(i,l)$.
• Proving that the derived category mutation corresponds exactly to the chord mutation under the $\Phi$-map.

Experimental results

Research questions

• RQ1 Is there a combinatorial invariant that classifies complete exceptional sequences in the derived category of $A_n$ up to shift and permutation?
• RQ2 How does the mutation of exceptional sequences correspond to a geometric or combinatorial transformation?
• RQ3 Can the number of such sequences be explicitly counted using known combinatorial objects?
• RQ4 What is the precise correspondence between indecomposable objects in the derived category and chords on a circle?

Key findings

• There exists a canonical bijection between the set of complete exceptional sequences modulo shift and permutation and the set of non-crossing spanning trees on $n+1$ points.
• The number of such equivalence classes is $\frac{1}{2n+1}\binom{3n}{n}$, matching the number of non-crossing spanning trees.
• Mutation of an exceptional pair $(E,F)$ corresponds to the chord mutation $\mathcal{L}_{\Phi(E)}\Phi(F) = \Phi(\mathcal{L}_E F)$.
• The derived category structure is fully encoded in the chord diagram via the $\Phi$-map, which preserves homological data.
• The classification is independent of the quiver orientation due to derived equivalence of $A_n$ path algebras.
• The result provides a complete, explicit, and combinatorially transparent description of all complete exceptional sequences over $A_n$.