Conway–Coxeter Friezes and Mutation: A Survey

Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko, Gordana Todorov

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

In this survey chapter, we explain the intricate links between Conway–Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.

Original languageEnglish
Title of host publicationAssociation for Women in Mathematics Series
Pages47-68
Number of pages22
DOIs
StatePublished - 2018

Publication series

NameAssociation for Women in Mathematics Series
Volume15
ISSN (Print)2364-5733
ISSN (Electronic)2364-5741

Bibliographical note

Funding Information:
We thank AWM for encouraging us to write this summary and giving us opportunity to continue this work. We also thank the referees for useful comments on the paper. EF, KS and GT received support from the AWM Advance grant to attend the symposium. KB was supported by the FWF grant W1230. KS was supported by NSF Postdoctoral Fellowship MSPRF-1502881.

Funding Information:
Acknowledgments We thank AWM for encouraging us to write this summary and giving us opportunity to continue this work. We also thank the referees for useful comments on the paper. EF, KS and GT received support from the AWM Advance grant to attend the symposium. KB was supported by the FWF grant W1230. KS was supported by NSF Postdoctoral Fellowship MSPRF-1502881.

Publisher Copyright:
© 2018, The Author(s) and the Association for Women in Mathematics.

Keywords

  • AR quiver
  • Caldero–Chapoton map
  • Cluster category
  • Cluster mutation
  • Cluster-tilted algebra
  • Frieze pattern
  • String module

ASJC Scopus subject areas

  • Mathematics (all)
  • Gender Studies

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