Conway-Coxeter friezes and beyond: Polynomially weighted walks around dissected polygons and generalized frieze patterns

[摘要] Conway and Coxeter introduced frieze patterns in 1973 and classified them via triangulated polygons. The determinant of the matrix associated to a frieze table was computed explicitly by Broline, Crowe and Isaacs in 1974, a result generalized 2012 by Baur and Marsh in the context of cluster algebras of type A. Higher angulations of polygons and associated generalized frieze patterns were studied in a joint paper with Holm and Jorgensen. Here we take these results further; we allow arbitrary dissections and introduce polynomially weighted walks around such dissected polygons. The corresponding generalized frieze table satisfies a complementary symmetry condition; its determinant is a multisymmetric multivariate polynomial that is given explicitly. But even more, the frieze matrix may be transformed over a ring of Laurent polynomials to a nice diagonal form generalizing the Smith normal form result given in [3]. Considering the generalized polynomial frieze in this context it is also shown that the non-zero local determinants are monomials that are given explicitly, depending on the geometry of the dissected polygon. (C) 2015 Elsevier Inc. All rights reserved.

[发布日期] 2015-11-15

[效力级别]  Proceedings Paper

[关键词] Frieze pattern;Polygon dissection;Weight matrix;Determinant;Diagonal form of a matrix;Polynomials