FRIEZE PATTERNS WITH COEFFICIENTS
FRIEZE PATTERNS WITH COEFFICIENTS
复制标题
带系数的饰带图案
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Peter Jørgensen
中科院分区：
文献类型：
--
作者：
M. Cuntz;T. Holm;Peter Jørgensen
Frieze模式，由Coxeter在1970年代引入，与无系数的簇代数密切相关。一个适当的推广frieze模式，链接到集群代数的系数，只简要地出现在一个未发表的手稿Propp。本文系统地研究了这些带系数的中楣图案，证明了一些基本结果，推广了中楣图案的经典结果。作为一个结果，我们看到如何frieze模式的系数，可以从经典的frieze模式，通过削减子多边形的三角形与经典的康威-考克斯特frieze模式。我们解决的问题，其中frieze模式的系数可以通过这种方式获得，并完全解决这个问题的三角形。最后，我们证明了有限性结果的frieze模式的系数表明，对于一个给定的边界序列，只有200多个（非零）frieze模式的系数与条目中的一个子集的复数没有一个聚点。
Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.