arXiv subjects

Liheng Tang

Publications and source records attributed to Liheng Tang.

2 recordsLinked to original sources

arXiv2026

$\mathrm{X}$-evolution flows/foliations for cluster algebras: partial Bongartz (co-)completions and acyclicity

We first generalize the notion of $\mathrm{X}$-foliation on cluster complexes to the context of cluster algebras. This is a continuous generalization of Keller's green mutation, which was previously introduced in the prequel within a categorical setting. We show that the wall-crossing structure (of possibly higher co-dimension) is encoded by $\mathrm{X}$-supports, which form a partial version of Cao-Gyoda-Yurikusa's Bongartz (co-)completions for cluster algebras. Moreover, we prove that for any cell transversed nontrivially by a fixed $\mathrm{X}$-leaf, its rank is an invariant bounded below by the rank of $\mathrm{X}$. Finally, we prove the acyclicity conjecture for $\mathrm{X}$-foliation, which also implies the compactness conjecture for any $\mathrm{X}$-foliation in finite type cases.

math.RT↗

arXiv2025

From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes

For any point $\mathrm{X}$ in the cluster complex $\mathrm{Cpx}(\mathcal{C})$ of a 2-Calabi-Yau category $\mathcal{C}$, we introduce $\mathrm{X}$-evolution flow on $\mathrm{Cpx}(\mathcal{C})$. We show that such a flow induces a piecewise linear one-dimensional $\mathrm{X}$-foliation with two singularities, the unique sink $\mathrm{X}$ and the unique source $\mathrm{X}[1]$. Moreover, we show that evolution flows on cluster complexes are continuous refinement/generalization of green mutations on cluster exchange graphs. For the cluster category of a Dynkin or Euclidean quiver $Q$, we prove that the $\mathrm{X}$-foliation is compact or semi-compact, for various choices of $\mathrm{X}$. As an application, we show that $\mathrm{Cpx}(\mathcal{C})$ is spherical (Dynkin case) or contractible (Euclidean case). As a byproduct, we show that the fundamental group of the cluster exchange graph of $Q$ is generated by squares and pentagons.

math.RT↗