[Paper Review] Extending Upper Cluster Algebras

Jiarui Fei, Jerzy Weyman|arXiv (Cornell University)|Jul 14, 2017

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper establishes conditions under which a ring extension of an upper cluster algebra remains an upper cluster algebra, using quiver mutation and potential theory. It proves that if a vertex can be optimized in a quiver with potential, and the original system is a cluster model, then the extended system is also a cluster model. The key result shows that semi-invariant rings of $m$-tuple flag quivers are upper cluster algebras with explicitly described quivers and rigid potentials.

ABSTRACT

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm 1}]$. We try to understand under which conditions ${R}$ is an upper cluster algebra, and how the quiver of $R$ relates to that of $S$. Moreover, if the restriction of $(Δ,W)$ to some subquiver is a cluster model, we give a sufficient condition for $(Δ,W)$ itself being a cluster model. As an application, we show that the semi-invariant ring of any complete $m$-tuple flags is an upper cluster algebra whose quiver is explicitly given. Moreover, the quiver with its rigid potential is a polyhedral cluster model.

Motivation & Objective

• To determine when a ring extension $ R $ of an upper cluster algebra $ S $, with $ R_{x_e} = S[x_e^{/pm 1}] $, is itself an upper cluster algebra.
• To characterize the quiver and seed structure of $ R $ in terms of $ S $ and the extended variable $ x_e $.
• To establish sufficient conditions under which an extended ice quiver with potential becomes a cluster model when the original is.
• To apply the theory to prove that semi-invariant rings of $ m $-tuple flag quivers are upper cluster algebras with explicit cluster structures.

Proposed method

• Use quiver mutation and the concept of 'optimizable' vertices to extend cluster models from subquivers to larger quivers with potential.
• Apply the covering pair technique and admissible mutation sequences to ensure compatibility between the original and extended cluster models.
• Employ algebraic geometry: if the common zero locus of two extended cluster variables has codimension 2 in $ \operatorname{Spec} R $, then $ R $ equals the upper cluster algebra.
• Construct a new seed $ (\Diamond_l(\mathbb{T}), \mathbb{s}_l(\mathbb{T})) $ by gluing smaller cluster seeds along triangulations of a disk with $ m $ marked points.
• Use mutation sequences corresponding to diagonal flips in ideal triangulations to relate different cluster seeds and show they generate the same upper cluster algebra.
• Prove that the resulting quiver with potential is a polyhedral cluster model by verifying the generic cluster character maps $ \mu $-supported $ \mathbf{g} $-vectors onto a basis.

Experimental results

Research questions

• RQ1 Under what conditions is a ring $ R $, locally a Laurent extension of an upper cluster algebra $ S $ via a cluster variable $ x_e $, itself an upper cluster algebra?
• RQ2 How does the quiver and seed of $ R $ relate to that of $ S $ when $ R_{x_e} = S[x_e^{\pm 1}] $?
• RQ3 When does an extended ice quiver with potential $ (\underline{\Delta}, \underline{W}) $ become a cluster model if the restriction $ (\Delta, W) $ is?
• RQ4 Can the semi-invariant ring of an $ m $-tuple flag quiver be realized as an upper cluster algebra with an explicit quiver and potential?
• RQ5 Is the cluster model structure preserved under gluing cluster seeds along triangulations of a disk with $ m $ marked points?

Key findings

• If each vertex in a set $ \boldsymbol{e} $ can be optimized in an ice quiver with potential $ (\underline{\Delta}, \underline{W}) $, and $ (\Delta, W) $ is a cluster model, then $ (\underline{\Delta}, \underline{W}) $ is also a cluster model.
• The semi-invariant ring $ \operatorname{SI}_{\beta_l}(S_l^m) $ of any complete $ m $-tuple flag quiver is isomorphic to an upper cluster algebra $ \overline{\mathcal{C}}(\Diamond_l(\mathbb{T}), \mathbb{s}_l(\mathbb{T})) $ for any ideal triangulation $ \mathbb{T} $ of a disk with $ m $ marked points.
• The quiver $ \Diamond_l(\mathbb{T}) $ with its rigid potential $ W_l(\mathbb{T}) $ forms a polyhedral cluster model, meaning the generic cluster character gives a basis of the upper cluster algebra.
• The cluster seeds associated with different triangulations $ \mathbb{T} $ and $ \mathbb{T}^\dagger $ (with no shared diagonals) yield the same upper cluster algebra, and the common zero locus of their respective cluster variables has codimension 2 in $ \operatorname{Spec}(\operatorname{SI}_{\beta_l}(S_l^m)) $.
• Mutation sequences corresponding to diagonal flips in triangulations preserve the cluster algebra structure, and the cluster variables transform via sign-preserving mutations.
• The proof of the main result is independent of the Knutson-Tao hive model, offering a simpler and more conceptual alternative to prior constructions.