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Theta functions in acyclic affine type

Nathan Reading, Salvatore Stella

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.

Theta functions in acyclic affine type

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.
Paper Structure (29 sections, 60 theorems, 46 equations, 3 figures)

This paper contains 29 sections, 60 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Basic background
  4. Scattering diagrams and theta functions
  5. The choice of coefficients
  6. The cluster monomials and $\mathbf{g}$-vectors
  7. Mutation of cluster scattering diagrams and theta functions
  8. Acyclic affine type
  9. The affine generalized associahedron fan
  10. The isomorphism $\nu_c$
  11. The affine cluster scattering diagram and fan
  12. Generalized cluster algebras
  13. Main Results
  14. Theta functions in the imaginary wall
  15. Imaginary exchange relations
  16. ...and 14 more sections

Key Result

Proposition 2.2

Suppose ${\tilde{B}}$ has linearly independent columns and $p_1,p_2\in P$. Then for every $\lambda\in P$, the formal power series $a(p_1,p_2,\lambda)$ does not depend on the sequence of points $\chi$ approaching $\lambda$. Furthermore,

Figures (3)

  • Figure 1: Defining the generalized seed
  • Figure 2: Seeds for $J_o$ and $J_o'=(J_o\setminus{\lbrace \gamma \rbrace})\cup\gamma'$ for $\gamma$ maximal
  • Figure 3: Seeds for $J_o$ and $J_o'=(J_o\setminus{\lbrace \gamma \rbrace})\cup\gamma'$ for $\gamma$ not maximal

Theorems & Definitions (88)

  • Remark 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Theorem 2.9
  • ...and 78 more