[Paper Review] Generalized cluster complexes via quiver representations

Bin Zhu|ArXiv.org|Jul 6, 2006

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper provides a quiver representation-theoretic interpretation of generalized cluster complexes using $d$-cluster categories, introducing a $d$-compatibility degree on colored almost positive real Schur roots. It establishes that the resulting simplicial complex, built from $d$-compatible roots, is isomorphic to the generalized cluster complex defined by Fomin and Reading, extending their construction to infinite root systems and unifying it with representation theory via $d$-cluster tilting objects.

ABSTRACT

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a $d-$compatibility degree $(-||-)$ on any pair of ``colored'' almost positive real Schur roots which generalizes previous definitions on the non-colored case, and call two such roots compatible provided the $d-$compatibility degree of them is zero. Associated to the root system $Φ$ corresponding to the valued quiver, by using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and $d-$compatible subsets as simplicies. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.

Motivation & Objective

• To provide a quiver representation-theoretic interpretation of generalized cluster complexes defined by Fomin and Reading.
• To extend the definition of generalized cluster complexes to infinite root systems, addressing a question posed by Fomin and Reading.
• To define a $d$-compatibility degree on colored almost positive real Schur roots using $d$-cluster categories.
• To establish a simplicial complex from $d$-compatible roots that is isomorphic to the cluster complex of the $d$-cluster category.
• To generalize the theory of cluster tilting objects and their combinatorics to arbitrary root systems via representation theory.

Proposed method

• The paper uses $d$-cluster categories, defined as triangulated orbit categories $\mathcal{D}/\tau^{-1}[d]$ of derived categories of representations of valued quivers.
• It introduces a $d$-compatibility degree $(-||-)$ between colored almost positive real Schur roots, generalizing the non-colored case.
• The compatibility is defined via the condition that the direct sum of corresponding exceptional objects in the $d$-cluster category is exceptional.
• It constructs a simplicial complex with colored almost positive real Schur roots as vertices and $d$-compatible subsets as simplices.
• It proves that this complex is isomorphic to the cluster complex of the $d$-cluster category, using a map $\gamma^d_\mathcal{H}$ that preserves faces and vertices.
• It generalizes BGP reflection functors to $d$-cluster categories and proves that any basic $d$-cluster tilting object has exactly $n$ indecomposable summands.

Experimental results

Research questions

• RQ1 Can generalized cluster complexes be interpreted via quiver representations and $d$-cluster categories?
• RQ2 Is it possible to extend the definition of generalized cluster complexes to infinite root systems?
• RQ3 What is the $d$-compatibility degree between colored almost positive real Schur roots in a $d$-cluster category?
• RQ4 How does the $d$-cluster category realize the generalized cluster complex combinatorially?
• RQ5 Are the generalized cluster complexes associated to different orientations of a Dynkin diagram isomorphic?

Key findings

• The generalized cluster complex $\Delta^{d,\mathcal{H}}(\Phi)$ is isomorphic to the cluster complex of the $d$-cluster category $\mathcal{C}_d(\mathcal{H})$ via the map $\gamma^d_\mathcal{H}$, which preserves vertices and faces.
• For any root system $\Phi$, the complex $\Delta^{d,\mathcal{H}}(\Phi)$ is pure of dimension $n-1$, generalizing a result of Fomin and Reading to infinite root systems.
• When $\Phi$ is a finite root system and $\mathcal{H}_0$ is the category of representations of an alternating quiver, $\Delta^{d,\mathcal{H}_0}(\Phi)$ coincides with the original generalized cluster complex $\Delta^d(\Phi)$ defined by Fomin and Reading.
• The number of $d$-cluster tilting objects in $\mathcal{C}_d(\mathcal{H})$ is $\prod_{i=1}^n \frac{dh + e_i + 1}{e_i + 1}$, where $h$ is the Coxeter number and $e_i$ the exponents of $\Phi$, generalizing known results for $d=1$.
• The number of complements of any almost complete $d$-cluster tilting object in $\mathcal{C}_d(\mathcal{H})$ is $d+1$, consistent with Thomas's result for simply-laced Dynkin types.