Cluster Monomials in Graph Laurent Phenomenon Algebras

TL;DR

This work shows that cluster monomials form a linear basis for graph Laurent phenomenon algebras, a class of LP algebras defined by directed graphs, thereby extending Caldero–Keller-type basis results from cluster algebras to a broader setting. The authors first prove linear independence by a triangularity argument linking cluster monomials to Laurent monomials with a diagonal, nonzero coefficient, then establish that cluster monomials span the algebra by showing any monomial (notably Y-monomials) can be expressed as a combination of cluster monomials; a key mechanism is the positive expansion of Y_IY_J over families of disjoint paths. A separate nonnegativity result is proved for bidirected trees, where every monomial expands into a nonnegative combination of cluster monomials, derived through explicit path-based decompositions and nwt-preserving bijections. Overall, the paper provides a robust foundation for the structural role of cluster monomials in graph LP algebras and identifies a tractable positivity regime on trees, with potential implications for combinatorial and representation-theoretic aspects of these algebras.

Abstract

Laurent phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Graph Laurent phenomenon algebras, defined by Lam and Pylyavskyy, are a subclass of Laurent phenomenon algebras whose structure is given by the data of a directed graph. In this paper, we prove that the cluster monomials of a graph Laurent phenomenon algebra form a linear basis, as conjectured by Lam and Pylyavskyy and analogous to a result for cluster algebras by Caldero and Keller. We also prove that, if the graph is a bidirected tree, the coefficients of the expansion of any monomial in terms of cluster monomials are nonnegative.

Figures (7)

• Figure 1: Two directed graphs. An undirected edge represents two directed edges, one in each direction. An undirected graph will mean a graph in which all edges are undirected.
• Figure 2: Multifunctions of the directed graph in Figure \ref{['fig:directedgraphs:1']}.
• Figure 3: The families of vertex-disjoint simple directed cycles in $\mathcal{C}_S$ for $S = \{1, 2, 3, 4\}$ and $\varGamma$ as in Figure \ref{['fig:directedgraphs:1']}. In order, the set $V(C)$ of vertices in each family $C$ is $\emptyset$, $\{1, 2, 3\}$, $\{1, 2, 3\}$, $\{1, 3, 4\}$, $\{1, 3, 4\}$, $\{1, 2, 3, 4\}$, $\{1, 2, 3, 4\}$, $\{1, 2\}$, $\{1, 3\}$, $\{1, 4\}$, $\{2, 3\}$, $\{3, 4\}$, $\{1, 2, 3, 4\}$, $\{1, 2, 3, 4\}$.
• Figure 4: Illustration of a family $\mathcal{P}$ of (two) disjoint paths from $I$ to $J$. The vertices of the blue region are in $W_I(\mathcal{P})$, the vertices of the red region are in $W_J(\mathcal{P})$, the vertices of blue or red regions are in $W_{I \cup J}(\mathcal{P})$, and the vertices of blue and red regions are in $W_{I \cap J}(\mathcal{P})$.
• Figure 5: Two examples of trees.

Theorems & Definitions (59)

• Theorem 1.1
• Conjecture 1.2: linearLP
• Theorem 1.3
• Lemma 2.1
• proof : Proof (surjectivity)
• proof : Proof (injectivity)
• Lemma 2.2
• proof
• Lemma 2.3: linearLP
• Theorem 3.1