论文标题
虚弱的饰物和饰面模式决定因素
Weak friezes and frieze pattern determinants
论文作者
论文摘要
Coxeter在1970年代引入了Frieze模式，由于它们与Fomin-Zelevinsky的集群代数，最近引起了新的兴趣。饰面模式可以解释为对对角线的对角线的分配，该分子满足对角线的某些条件（托勒密关系）。正如Canakci和Jorgensen引入的那样，弱的饰边正在通过允许解剖的多边形粘合来推广这一概念，以便只能满足涉及其中一个粘合对角线的交叉点。对于任何frize模式，人们都可以使用矩阵上和下半部分的三角形基本结构域将对称矩阵与对角线相关联。 Broline，Crowe和Isaacs已经找到了这些矩阵决定因素的公式，后来他们的工作已被其他作者在各个方向上概括。这些饰面模式决定因素是我们论文的主要重点。作为我们的主要结果，我们表明这种决定因素在粘合弱的饰边方面表现良好：决定因素是粘合的碎片的决定因素的乘积，最多来自粘合对角线的标量因子。然后，我们给出了该结果的几个应用，表明文献中的公式是由Broline-Crowe-Isaacs，Baur-Marsh，Bessenrodt-Holm-Jorgensen和Maldonado获得的。
Frieze patterns have been introduced by Coxeter in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Canakci and Jorgensen, are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh, Bessenrodt-Holm-Jorgensen and Maldonado can all be obtained as consequences of our result.
