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Y-frieze patterns

Antoine de Saint Germain

Abstract

Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call ${\bf Y}\textit{-frieze patterns}$. Using the mutation rules for ${\bf Y}$-variables, we define a large class of ${\bf Y}$-frieze patterns called $\textit{unitary }{\bf Y}\textit{-frieze patterns}$, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) ${\bf Y}$-frieze patterns. In rank 2, we show that ${\bf Y}$-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that ${\bf Y}$-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite.

Y-frieze patterns

Abstract

Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call . Using the mutation rules for -variables, we define a large class of -frieze patterns called , and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) -frieze patterns. In rank 2, we show that -frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that -frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite.
Paper Structure (20 sections, 18 theorems, 99 equations, 4 figures)

This paper contains 20 sections, 18 theorems, 99 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Acknowledgements
  5. Notation
  6. Y-frieze patterns and their properties
  7. Semi-rings and semi-fields
  8. Frieze patterns and Y-frieze patterns
  9. Constructing Y-frieze patterns from frieze patterns
  10. Background on matrix, A and Y-patterns
  11. Unitary Y-frieze patterns
  12. The Y-cluster structure of Y-frieze patterns
  13. Friezes of positive spaces
  14. Friezes and Y-frieze patterns
  15. Y-frieze patterns in rank 2
  16. ...and 5 more sections

Key Result

Lemma 2.7

Let $A$ be any $r \times r$ symmetrisable generalised Cartan matrix, $S$ any semi-field. One has bijections Given ${\bf s} = (s_1, \ldots, s_r) \in S^r$, the unique $f\in {\rm Frieze}(A,S), k \in {\rm YFrieze}(A,S)$ such that are denoted $f = f_{\bf s}$ and $k = k_{\bf s}$ respectively. The procedure, recursive with respect to the total order eq:tot-order-grid, for computing the values of $f_{\b

Figures (4)

  • Figure 1: Arithmetic Y-frieze patterns. In type $A_r$, one obtains a Y-frieze pattern of width $r$ (c.f. \ref{['eq:Y-frieze-pattern']}) by adding a top and bottom row of $0$'s.
  • Figure 2: The generic Y-frieze pattern of type $A_3$.
  • Figure 3: Expression of $y(i,m)'s$ as Laurent polynomials in the initial Y-cluster in type $A_3$.
  • Figure 4: The superunitary region (embedded in ${\mathbb R}^2$) in types $A_2, C_2$ and $G_2$.

Theorems & Definitions (49)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Theorem 2.8
  • proof
  • Example 2.9
  • ...and 39 more