Details
| Original language | English |
|---|---|
| Pages (from-to) | 80-103 |
| Number of pages | 24 |
| Journal | Journal of algebra |
| Volume | 442 |
| Early online date | 20 Mar 2015 |
| Publication status | Published - 15 Nov 2015 |
Abstract
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.
Keywords
- Determinant, Diagonal form of a matrix, Frieze pattern, Polygon dissection, Polynomials, Weight matrix
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 442, 15.11.2015, p. 80-103.
Research output: Contribution to journal › Article › Research › peer review
}
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 -