Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing

Abstract

This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a $T$-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Canakci and Jorgensen.

Figures (14)

• Figure 1: A Coxeter frieze pattern. There are borders of zeroes and ones, and the remaining entries are positive integers such that each adjacent $2 \times 2$-matrix has determinant one.
• Figure 2: Notation for the entries of a frieze pattern.
• Figure 3: The friezes of Definition \ref{['def:frieze']} satisfy the triangle relation $c_{ ij }c_{ kj }^{ -1 }c_{ ki } = c_{ ik }c_{ jk }^{ -1 }c_{ ji }$ (left) and the exchange relation $c_{ ik } = c_{ ij }c_{ \ell j }^{ -1 }c_{ \ell k } + c_{ i \ell }c_{ j \ell }^{ -1 }c_{ jk }$ (right).
• Figure 4: A frieze pattern with values in the quaternions. Reproduced from Cuntz-Holm-Jorgensen-noncomm.
• Figure 5: The square matrix $M_c$ arising from a map $c : \operatorname{diag} P \xrightarrow{} R$ where $P$ is an $n$-gon with vertex set numbered by $\{\, 1, \ldots, n \,\}$. Theorem \ref{['thm:A']} computes $\det M_c$ when $c$ is a frieze, $R$ a skew field.

Theorems & Definitions (29)

• Definition 1: Friezes
• Theorem A: Non-commutative frieze determinants
• Definition 2: Weak friezes
• Theorem B: The non-commutative Laurent phenomenon
• Theorem C: Gluing weak friezes
• Theorem D: Gluing friezes
• Definition 1.2: Polygons
• Definition 1.3: $T$-paths
• Definition 1.4: The notation $\mathscr{P}$
• Definition 1.5: The $T$-path formula