Exceptional sequences and rooted labeled forests

TL;DR

This work provides a representation-theoretic combinatorial framework for complete exceptional sequences of type $A_n$ by establishing a bijection with rooted labeled forests. Through this lens, relatively projective and relatively injective objects are read directly from the forest topology, and a three-variable generating function counts sequences by these types. The authors equip rooted labeled forests with a natural braid-group action that mirrors the action on exceptional sequences, and they analyze the Garside element $\Delta$ to illuminate connections with cluster theory and signed exceptional sequences. They also relate parking-function constructions and chord diagrams to their forest model, yielding a versatile visualization and counting toolkit with potential extensions to cluster-tilting theory and related algebras.

Abstract

We give a representation-theoretic bijection between rooted labeled forests with $n$ vertices and complete exceptional sequences for the quiver of type $A_n$ with straight orientation. The ascending and descending vertices in the forest correspond to relatively injective and relatively projective objects in the exceptional sequence. We conclude that every object in an exceptional sequence for linearly oriented $A_n$ is either relatively projective or relatively injective or both. We construct a natural action of the extended braid group on rooted labeled forests and show that it agrees with the known action of the braid group on complete exceptional sequences. We also describe the action of $Δ$, the Garside element of the braid group, on rooted labeled forests using representation theory and show how this relates to cluster theory.

Figures (13)

• Figure 1: By Theorem \ref{['thm A1']}, this figure indicates the rooted labeled forest corresponding to the complete exceptional sequence for the quiver $A_7$: $(E_1,E_2,E_3,E_4,E_5,E_6,E_7) = (M_{33} , M_{13} , M_{11} , M_{66} , M_{77} , M_{47} , M_{44})\qquad\qquad\qquad$$M_{ab}$ denotes the module with support $[a,b]$. E.g., $M_{13}$ contains $M_{11}$ and $M_{33}$ in its support and $M_{aa}=S_a$ is the simple module at $a$.
• Figure 2: The exceptional sequence $(M_{23},S_6,M_{13}, M_{79}, M_{49}, S_4, S_7, S_2, S_8)$ from Example \ref{['eg: rooted forest for A9']} is drawn as the sequence of chords $(24,67,14,70,40,45,78,23,89)$ in red. The corresponding tree with root $(\ast)$ in the region containing the arc $0$ to $1$ is indicated in blue. Deleting the root gives the rooted labeled forest in Example \ref{['eg: rooted forest for A9']}.
• Figure 3: Cases 1,2,3 of the braid move $\sigma_i$ are indicated. $v_k$ denotes the smallest node in $F_+$ which is above both $v_i$ and $v_{i+1}$ and $Z$ denotes the set of other children of $v_k$. $\sigma_i^3$ is the identity in all three cases.
• Figure 4: Braid moves $\sigma_5,\sigma_3,\sigma_2,\sigma_4$ illustrated above are examples of Cases a,b,c,d in the proof of Theorem \ref{['thm: action of sigma-i corresponds']}.
• Figure 5: The three rooted labeled forests on $n=2$ are cyclically permuted by $\sigma_1$.

Theorems & Definitions (92)

• Theorem A1
• Theorem A2
• Corollary B
• Theorem C
• Definition 1.1
• Remark 1.2
• Lemma 1.3
• proof
• Lemma 1.4
• proof