Theta functions in acyclic affine type

TL;DR

This paper characterizes theta functions associated with the imaginary wall in acyclic affine type cluster algebras, revealing their algebraic structure and new exchange relations, and decomposing the imaginary subalgebra into tensor products of generalized cluster algebras.

Contribution

It provides a detailed characterization of theta functions in the imaginary wall and introduces the concept of the imaginary subalgebra with its tensor product decomposition.

Findings

Theta functions span the imaginary subalgebra of the cluster algebra.

The imaginary subalgebra decomposes into tensor products of tube subalgebras.

New imaginary exchange relations among cluster variables are identified.

Abstract

We characterize the theta functions for vectors in the imaginary wall in a cluster algebra of acyclic affine type and compute some of their structure constants. One of the structure constant computations can be interpreted as new "imaginary" exchange relations among cluster variables. We show that theta functions in the imaginary wall span a subalgebra of the cluster algebra that we call the imaginary subalgebra, which decomposes as a tensor product of tube subalgebras that are generalized cluster algebras of type C. Our proofs exploit mutation-symmetries of the exchange matrix, an earlier characterization of dominance regions in affine type, and combinatorial models for cluster scattering diagrams of acyclic affine type.