math.QA — Quantum Algebra

Finite quantum theta functions for skew-symmetrizable seeds

Contributed by 琪越 唐

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.