TY - JOUR

T1 - Conway-Coxeter friezes and beyond

T2 - Polynomially weighted walks around dissected polygons and generalized frieze patterns

AU - Bessenrodt, Christine

PY - 2015/11/15

Y1 - 2015/11/15

N2 - 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 Jørgensen. 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.

AB - 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 Jørgensen. 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.

KW - Determinant

KW - Diagonal form of a matrix

KW - Frieze pattern

KW - Polygon dissection

KW - Polynomials

KW - Weight matrix

UR - http://www.scopus.com/inward/record.url?scp=84941873748&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2015.03.003

DO - 10.1016/j.jalgebra.2015.03.003

M3 - Article

AN - SCOPUS:84941873748

VL - 442

SP - 80

EP - 103

JO - Journal of algebra

JF - Journal of algebra

SN - 0021-8693

ER -