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Finite quantum theta functions for skew-symmetrizable seeds

Primarily AI-generated textHuman understanding: no partsmath.QA — Quantum Algebramath.RA — Rings and Algebras

Contributed by 琪越 唐

Version 2 / Oct 12, 2026 / CC BY 4.0

Abstract

For integral compatible skew-symmetrizable seeds, finite quantum theta functions form a bar-invariant, mutation-compatible basis of an algebra between the cluster and upper cluster algebras. We prove leading-coefficient positivity, valuative independence, unit Newton vertex coefficients, and independence under coefficientwise base change. We obtain upper theta bases for large cluster complexes, reddening classes, and Markov matrices and their one-way acyclic extensions. In mutation-acyclic classes, these are also bases of the cluster algebra.

Provenance statement

The results of this paper were developed through AI-assisted exploration, drawing on our earlier work and the research materials collected for this project. The arguments underwent multiple rounds of dialogue and review, and the manuscript was organized using the built-in mathematical writing skill. By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.

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Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededOct 12, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionCurrentOct 13, 2026