Details
| Original language | English |
|---|---|
| Pages (from-to) | 194-245 |
| Number of pages | 52 |
| Journal | Advances in mathematics |
| Volume | 315 |
| Early online date | 13 Jun 2017 |
| Publication status | Published - 31 Jul 2017 |
Abstract
An SL2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2×2-submatrix has determinant 1. Such tilings are infinite analogues of Conway–Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway–Coxeter counting, every SL2-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered SL2-tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.
Keywords
- Conway–Coxeter frieze, Disc with accumulation points, Igusa–Todorov cluster category, Ptolemy formula, Tiling, Triangulation
ASJC Scopus subject areas
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In: Advances in mathematics, Vol. 315, 31.07.2017, p. 194-245.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - All SL2-tilings come from infinite triangulations
AU - Bessenrodt, Christine
AU - Holm, Thorsten
AU - Jørgensen, Peter
N1 - Funding information: This work was carried out while Peter Jørgensen was visiting the Leibniz Universität Hannover. He thanks Thorsten Holm and the Institut für Algebra, Zahlentheorie und Diskrete Mathematik for their hospitality. He also gratefully acknowledges financial support from Thorsten Holm's grant HO 1880/5-1, which is part of the research priority programme SPP 1388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft (DFG).
PY - 2017/7/31
Y1 - 2017/7/31
N2 - An SL2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2×2-submatrix has determinant 1. Such tilings are infinite analogues of Conway–Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway–Coxeter counting, every SL2-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered SL2-tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.
AB - An SL2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2×2-submatrix has determinant 1. Such tilings are infinite analogues of Conway–Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway–Coxeter counting, every SL2-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered SL2-tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.
KW - Conway–Coxeter frieze
KW - Disc with accumulation points
KW - Igusa–Todorov cluster category
KW - Ptolemy formula
KW - Tiling
KW - Triangulation
UR - http://www.scopus.com/inward/record.url?scp=85019974106&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2017.05.019
DO - 10.1016/j.aim.2017.05.019
M3 - Article
AN - SCOPUS:85019974106
VL - 315
SP - 194
EP - 245
JO - Advances in mathematics
JF - Advances in mathematics
SN - 0001-8708
ER -