On Christoffel words & their lexicographic array

TL;DR

This paper explores the algebraic properties of Christoffel matrices derived from Christoffel words, demonstrating their commutative multiplication, invertibility characteristics, and connections to group representations over finite fields.

Contribution

It establishes that Christoffel matrices form an abelian subgroup of invertible matrices and characterizes their structure over various rings and fields.

Findings

Product of Christoffel matrices is commutative over an integral domain.

Invertible Christoffel matrices have invertible Christoffel matrices as inverses.

Finite fields and groups can be faithfully represented by Christoffel matrices.

Abstract

By a Christoffel matrix we mean a $n\times n$ matrix corresponding to the lexicographic array of a Christoffel word of length $n.$ In this note we show that if $R$ is an integral domain, then the product of two Christoffel matrices over $R$ is commutative and is a Christoffel matrix over $R.$ Furthermore, if a Christoffel matrix over $R$ is invertible, then its inverse is a Christoffel matrix over $R.$ Consequently, the set $GC_n(R)$ of all $n\times n$ invertible Christoffel matrices over $R$ forms an abelian subgroup of $GL_n(R).$ The subset of $GC_n(R)$ consisting all invertible Christoffel matrices having some element $a$ on the diagonal and $b$ elsewhere (with $a,b \in R$ distinct) forms a subgroup $H$ of $GC_n(R).$ If $R$ is a field, then the quotient $GC_n(R)/H$ is isomorphic to $(\Z/nZ)^\times,$ the multiplicative group of integers modulo $n.$ It follows that for each finite field $F$ and each finite abelian group $G,$ there exists $n\geq 2$ and a faithful representation $G\rightarrow GL_n(F)$ consisting entirely of $n\times n$ (invertible) Christoffel matrices over $F.$ We describe the structure of $GC_n(\Z/2\Z).$