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A module-theoretic interpretation of quantum expansion formula

Yutong Yu

Abstract

We provide a module-theoretic interpretation of the expansion formula given by Huang (2022), which defines a map on perfect matchings to compute the expansion of quantum cluster variables in quantum cluster algebras arising from unpunctured surfaces. In addition, we present a multiplication formula for string modules with one-dimensional extension space, derived using the skein relations. For the Kronecker type, an alternative expansion formula was given in Canakci and Lampe (2020), and we show that the two expansion formulas coincide.

A module-theoretic interpretation of quantum expansion formula

Abstract

We provide a module-theoretic interpretation of the expansion formula given by Huang (2022), which defines a map on perfect matchings to compute the expansion of quantum cluster variables in quantum cluster algebras arising from unpunctured surfaces. In addition, we present a multiplication formula for string modules with one-dimensional extension space, derived using the skein relations. For the Kronecker type, an alternative expansion formula was given in Canakci and Lampe (2020), and we show that the two expansion formulas coincide.

Paper Structure

This paper contains 17 sections, 33 theorems, 50 equations, 12 figures.

Table of Contents

  1. INTRODUCTION
  2. REVIEW OF CLUSTER ALGEBRA
  3. GENTLE ALGEBRA FROM SURFACE
  4. CLUSTER EXPANSION FORMULA
  5. Snake graph and perfect matchings
  6. Quantum Laurent expansions
  7. Snake graphs and string modules
  8. MODULE INTERPRETATION
  9. THE SKEIN ALGEBRA
  10. A MULTIPLICATION FORMULA WITH ONE DIMENSIONAL EXTENSION SPACE
  11. Arrow extension
  12. Overlap extension
  13. The Kronecker Case
  14. Results in CANAKCI2020105132
  15. Equality
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a map $v_\gamma:CS(M(w))\rightarrow \mathbb{Z}$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (12)

  • Figure 1: Neighborhood of $\tau_k$
  • Figure 2: $a_{j-1}$ and $a_j$ both are inverse
  • Figure 3: $a_{j-1}$ and $a_j$ both are direct
  • Figure 4: $x_{j-1}\leftarrow x_{j}\rightarrow x_{j+1}$
  • Figure 5: $x_{j-1}\rightarrow x_{j}\leftarrow x_{j+1}$
  • ...and 7 more figures

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: BERENSTEIN2005405
  • Definition 3.1
  • Lemma 3.2: 10.2140/ant.2010.4.201
  • Remark 3.3
  • ...and 63 more