Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weighted Cycles on Weaves

Daping Weng

TL;DR

The paper develops a weighted-cycle framework for weaves across all Dynkin types, linking planar weave networks to cluster structures through merodromies on decorated flag moduli spaces. It defines the weighted cycle algebra $\mathcal{W}(\mathfrak{w})$, proves it is a Laurent polynomial algebra of rank $\tau+r(\beta-1)$, and constructs a quantization $\mathbb{W}(\mathfrak{w})$ in the simply-laced case via a skew-symmetrizable intersection pairing. Merodromies along weighted cycles realize cluster variables, and mutations of weighted cycles reproduce cluster mutations; the work also connects this framework to Demazure weaves, generalized minors, cross-ratios, triple-ratios, and quantum groups. Finally, it extends the construction to non-simply-laced types through foldings, multipliers, and coweight considerations, creating a broad, algebraic-topological bridge between webs, skein ideas, and cluster algebras across Dynkin types.

Abstract

We introduce weighted cycles on weaves of general Dynkin types and define a skew-symmetrizable intersection pairing between weighted cycles. We prove that weighted cycles on a weave form a Laurent polynomial algebra and construct a quantization for this algebra using the skew-symmetric intersection pairing in the simply-laced case. We define merodromies along weighted cycles as functions on the decorated flag moduli space of the weave. We relate weighted cycles with cluster variables in a cluster algebra and prove that mutations of weighted cycles are compatible with mutations of cluster variables.

Weighted Cycles on Weaves

TL;DR

The paper develops a weighted-cycle framework for weaves across all Dynkin types, linking planar weave networks to cluster structures through merodromies on decorated flag moduli spaces. It defines the weighted cycle algebra , proves it is a Laurent polynomial algebra of rank , and constructs a quantization in the simply-laced case via a skew-symmetrizable intersection pairing. Merodromies along weighted cycles realize cluster variables, and mutations of weighted cycles reproduce cluster mutations; the work also connects this framework to Demazure weaves, generalized minors, cross-ratios, triple-ratios, and quantum groups. Finally, it extends the construction to non-simply-laced types through foldings, multipliers, and coweight considerations, creating a broad, algebraic-topological bridge between webs, skein ideas, and cluster algebras across Dynkin types.

Abstract

We introduce weighted cycles on weaves of general Dynkin types and define a skew-symmetrizable intersection pairing between weighted cycles. We prove that weighted cycles on a weave form a Laurent polynomial algebra and construct a quantization for this algebra using the skew-symmetric intersection pairing in the simply-laced case. We define merodromies along weighted cycles as functions on the decorated flag moduli space of the weave. We relate weighted cycles with cluster variables in a cluster algebra and prove that mutations of weighted cycles are compatible with mutations of cluster variables.

Paper Structure

This paper contains 22 sections, 21 theorems, 72 equations, 42 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decorated Flags and
  4. Weaves and Y-cycles
  5. Weighted Cycles and Their Merodromies
  6. Definition of Weighted Cycles
  7. Intersection Pairing between Weighted Cycles
  8. Weighted Cycle Algebra and Quantization
  9. Weighted Cycle Representatives for Y-Cycles
  10. Merodromies
  11. Mutations
  12. Bivalent Weave Vertices
  13. Applications
  14. Cluster Theory
  15. Demazure Weaves
  16. ...and 7 more sections

Key Result

Lemma 2.3

Suppose $x^{-1}y\in Bs_iB$. Then $h_\pm(xN,yN)=(-1)^{\alpha_i^\vee}h_\mp(xN,yN)$.

Figures (42)

  • Figure 1: Left: an $\mathrm{SL}_3$-web. Right: the lift of this $\mathrm{SL}_3$-web to a weighted cycle.
  • Figure 2: Allowable vertices in a weave.
  • Figure 3: Intersection pairing between frozen Y-cycles.
  • Figure 4: Weave equivalences.
  • Figure 5: Mutation at a short I-cycle
  • ...and 37 more figures

Theorems & Definitions (88)

  • Conjecture 4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 78 more