{"title":"关于 Christoffel 词及其词典阵列","authors":"Luca Q. Zamboni","doi":"arxiv-2409.07974","DOIUrl":null,"url":null,"abstract":"By a Christoffel matrix we mean a $n\\times n$ matrix corresponding to the\nlexicographic array of a Christoffel word of length $n.$ If $R$ is an integral\ndomain, then the product of two Christoffel matrices over $R$ is commutative\nand is a Christoffel matrix over $R.$ Furthermore, if a Christoffel matrix over\n$R$ is invertible, then its inverse is a Christoffel matrix over $R.$\nConsequently, the set $GC_n(R)$ of all $n\\times n$ invertible Christoffel\nmatrices over $R$ forms an abelian subgroup of $GL_n(R).$ The subset of\n$GC_n(R)$ consisting all invertible Christoffel matrices having some element\n$a$ on the diagonal and $b$ elsewhere (with $a,b \\in R$ distinct) forms a\nsubgroup $H$ of $GC_n(R).$ If $R$ is a field, then the quotient $GC_n(R)/H$ is\nisomorphic to $(\\Z/nZ)^\\times,$ the multiplicative group of integers modulo\n$n.$ It follows from this that for each finite field $F$ and each finite\nabelian group $G,$ there exists $n\\geq 2$ and a faithful representation\n$G\\rightarrow GL_n(F)$ consisting entirely of $n\\times n$ (invertible)\nChristoffel matrices over $F.$ We find that $GC_n(\\F_2) \\simeq (\\Z/nZ)^\\times$\nfor $n$ odd and $GC_n(\\F_2) \\simeq \\Z/2\\Z \\times (\\Z/nZ)^\\times$ for $n$ even.\nAs an application, we define an associative and commutative binary operation on\nthe set of all $\\{0,1\\}$-Christoffel words of length $n$ which in turn induces\nan associative and commutative binary operation on $\\{0,1\\}$-central words of\nlength $n-2.$","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Christoffel words & their lexicographic array\",\"authors\":\"Luca Q. Zamboni\",\"doi\":\"arxiv-2409.07974\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"By a Christoffel matrix we mean a $n\\\\times n$ matrix corresponding to the\\nlexicographic array of a Christoffel word of length $n.$ If $R$ is an integral\\ndomain, then the product of two Christoffel matrices over $R$ is commutative\\nand is a Christoffel matrix over $R.$ Furthermore, if a Christoffel matrix over\\n$R$ is invertible, then its inverse is a Christoffel matrix over $R.$\\nConsequently, the set $GC_n(R)$ of all $n\\\\times n$ invertible Christoffel\\nmatrices over $R$ forms an abelian subgroup of $GL_n(R).$ The subset of\\n$GC_n(R)$ consisting all invertible Christoffel matrices having some element\\n$a$ on the diagonal and $b$ elsewhere (with $a,b \\\\in R$ distinct) forms a\\nsubgroup $H$ of $GC_n(R).$ If $R$ is a field, then the quotient $GC_n(R)/H$ is\\nisomorphic to $(\\\\Z/nZ)^\\\\times,$ the multiplicative group of integers modulo\\n$n.$ It follows from this that for each finite field $F$ and each finite\\nabelian group $G,$ there exists $n\\\\geq 2$ and a faithful representation\\n$G\\\\rightarrow GL_n(F)$ consisting entirely of $n\\\\times n$ (invertible)\\nChristoffel matrices over $F.$ We find that $GC_n(\\\\F_2) \\\\simeq (\\\\Z/nZ)^\\\\times$\\nfor $n$ odd and $GC_n(\\\\F_2) \\\\simeq \\\\Z/2\\\\Z \\\\times (\\\\Z/nZ)^\\\\times$ for $n$ even.\\nAs an application, we define an associative and commutative binary operation on\\nthe set of all $\\\\{0,1\\\\}$-Christoffel words of length $n$ which in turn induces\\nan associative and commutative binary operation on $\\\\{0,1\\\\}$-central words of\\nlength $n-2.$\",\"PeriodicalId\":501407,\"journal\":{\"name\":\"arXiv - MATH - Combinatorics\",\"volume\":\"27 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Combinatorics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.07974\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07974","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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 $n-2.$