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All SL₂ Tilings come from Infinite Triangulations

Lookup NU author(s): Professor Peter Jorgensen

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Abstract

Abstract. An SL2-tiling is a bi-innite matrix of positive integers such that each adjacent2 2-submatrix has determinant 1. Such tilings are innite analogues of Conway{Coxeterfriezes, 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 froma triangulation of the disc with two, three or four accumulation points.This improves earlier results which only discovered SL2-tilings with innitely many entriesequal to 1. Indeed, our methods show that there are large classes of tilings with only nitelymany entries equal to 1, including a class of tilings with no 1's at all. In the latter case, weshow that the minimal entry of a tiling is unique.

Publication metadata

Author(s): Bessenrodt C, Holm T, Jorgensen P

Publication type: Article

Publication status: Published

Journal: Advances in Mathematics

Year: 2017

Volume: 315

Pages: 194-245

Print publication date: 31/07/2017

Online publication date: 13/06/2017

Acceptance date: 23/05/2017

Date deposited: 24/05/2017

ISSN (print): 0001-8708

ISSN (electronic): 1090-2082

Publisher: Elsevier

URL: https://doi.org/10.1016/j.aim.2017.05.019

DOI: 10.1016/j.aim.2017.05.019

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