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 language | English |
---|---|
Title of host publication | Association for Women in Mathematics Series |
Pages | 47-68 |
Number of pages | 22 |
DOIs | |
State | Published - 2018 |
Publication series
Name | Association for Women in Mathematics Series |
---|---|
Volume | 15 |
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