On Christoffel words & their lexicographic array

Abstract

By a Christoffel matrix we mean a $n\times n$ matrix corresponding to the lexicographic array of a Christoffel word of length $n.$ 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 from this 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 find that $GC_n(\F_2) \simeq (\Z/nZ)^\times$ for $n$ odd and $GC_n(\F_2) \simeq \Z/2\Z \times (\Z/nZ)^\times$ for $n$ even. As an application, we define an associative and commutative binary operation on the set of all $\{0,1\}$-Christoffel words of length $n$ which in turn induces an associative and commutative binary operation on $\{0,1\}$-central words of length $n-2.$