Valuative independence and cluster theta reciprocity

TL;DR

This paper proves valuative independence and theta reciprocity for theta functions from positive scattering diagrams, leading to new insights into their linear independence, invariance properties, and applications to cluster varieties.

Contribution

It introduces a framework for valuative independence and theta reciprocity, providing new tools for understanding theta functions in cluster algebra theory.

Findings

Theta functions satisfy valuative independence for certain valuations.

Theta functions exhibit a symmetry called theta reciprocity.

Results enable identification of theta function bases for line bundles on cluster varieties.

Abstract

We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations $\operatorname{val}_{v}$, we have $\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \operatorname{val}_v(\vartheta_u)$. As applications, we prove linear independence of theta functions with specialized coefficients and characterize when theta functions for cluster varieties are unchanged by the unfreezing of an index. This yields a general gluing result for theta functions from moduli of local systems on marked surfaces. We then prove that theta functions for cluster varieties satisfy a symmetry property called theta reciprocity: briefly, $\operatorname{val}_v(\vartheta_u)=\operatorname{val}_u(\vartheta_v)$. For this we utilize a new framework called a "seed datum" for understanding cluster-type varieties. One may apply valuative independence and theta reciprocity together to identify theta function bases for global sections of line bundles on partial compactifications of cluster varieties.