关于 Christoffel 词及其词典阵列

Luca Q. Zamboni
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引用次数: 0

摘要

我们所说的 Christoffel 矩阵是指与长度为 $n 的 Christoffel 字的反射数组相对应的 $n/times n$ 矩阵。$ 如果 $R$ 是一个积分域,那么 $R$ 上的两个 Christoffel 矩阵的乘积是交换的,并且是 $R 上的 Christoffel 矩阵。$GC_n(R)$的子集构成$GC_n(R)的子群$H$.$ 如果 $R$ 是一个域,那么商$GC_n(R)/H$ 与$(\Z/nZ)^\times, $整数模的乘法群同构。由此可见,对于每个有限域 $F$ 和每个有限阿贝尔群 $G$ 都存在 $n\geq 2$ 和一个完全由 $n\times n$ (可反)Christoffel 矩阵组成的忠实表示 $G\rightarrow GL_n(F)$。我们发现 $n$ 奇数时,$GC_n(\F_2) \simeq (\Z/nZ)^\times$ 而 $n$ 偶数时,$GC_n(\F_2) \simeq \Z/2\Z \times (\Z/nZ)^\times$ 。作为一个应用,我们在长度为$n$的所有$\{0,1\}$-Christoffel词的集合上定义了一个联立和交换二元运算,这反过来又在长度为$n-2的$\{0,1\}$-中心词上引起了一个联立和交换二元运算。
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On Christoffel words & their lexicographic array
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 $n-2.$
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