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Christoffel Matrices and Sturmian Determinants

Christophe Reutenauer, Jeffrey Shallit

TL;DR

The paper analyzes Christoffel words via their Burrows-Wheeler (Christoffel) matrices, proving these matrices form a commutative group isomorphic to $K^*\times K^*\times G_n$ and giving explicit inverse and determinant formulas involving the Zolotarev symbol. It then studies Sturmian determinants: for a Sturmian sequence $g$, the determinantal vector $V_n$ derived from length-$n$ factors is shown to be a perfectly clustering word over two or three letters, encoding a symmetric discrete interval exchange, with detailed dependence on Christoffel word factorizations and continued fractions. Special cases include the Fibonacci word, where determinants take Fibonacci-number values (up to sign) with the sign described by the Zolotarev symbol; the work also connects to continued fractions, semi-convergents, and Stern–Brocot paths. Overall, the paper links combinatorics on words, linear-algebraic matrix structures, and number-theoretic signs to illuminate the structure of Sturmian determinants and their algebraic and geometric interpretations.

Abstract

We discuss certain matrices associated with Christoffel words, and show that they have a group structure. We compute their determinants and show a relationship between the Zolotareff symbol from number theory.

Christoffel Matrices and Sturmian Determinants

TL;DR

The paper analyzes Christoffel words via their Burrows-Wheeler (Christoffel) matrices, proving these matrices form a commutative group isomorphic to and giving explicit inverse and determinant formulas involving the Zolotarev symbol. It then studies Sturmian determinants: for a Sturmian sequence , the determinantal vector derived from length- factors is shown to be a perfectly clustering word over two or three letters, encoding a symmetric discrete interval exchange, with detailed dependence on Christoffel word factorizations and continued fractions. Special cases include the Fibonacci word, where determinants take Fibonacci-number values (up to sign) with the sign described by the Zolotarev symbol; the work also connects to continued fractions, semi-convergents, and Stern–Brocot paths. Overall, the paper links combinatorics on words, linear-algebraic matrix structures, and number-theoretic signs to illuminate the structure of Sturmian determinants and their algebraic and geometric interpretations.

Abstract

We discuss certain matrices associated with Christoffel words, and show that they have a group structure. We compute their determinants and show a relationship between the Zolotareff symbol from number theory.
Paper Structure (18 sections, 18 theorems, 60 equations, 1 figure)

This paper contains 18 sections, 18 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Group of Christoffel matrices
  3. Burrows-Wheeler matrix of a Christoffel word
  4. Group of Christoffel matrices
  5. Consequences
  6. Sturmian determinants
  7. Determinants of Sturmian factors
  8. Perfectly clustering words and symmetric discrete interval exchanges
  9. Factors of Sturmian sequences
  10. Determinants
  11. Proof of Theorem \ref{['main']}, first part
  12. Some preliminary results
  13. Proof of Theorem \ref{['main']}, second part, and Corollary \ref{['VV']}
  14. Example
  15. Continued fractions
  16. ...and 3 more sections

Key Result

Theorem 3.1

Suppose that the characteristic of $\mathbb K$ is $> n$. The set of all matrices $M_n(a,b,r)$, where $a,b\in \mathbb K$, $a\neq b$, $r\in[n]$, $r,n$ relatively prime, and $(n-r)a+rb\neq 0$, forms a commutative subgroup $\mathcal{G}$ of $\mathop{\mathrm{GL}}\nolimits_n(\mathbb K)$. The mapping $\math

Figures (1)

  • Figure 1: Burrows-Wheeler matrix of the Christoffel word $0001001$.

Theorems & Definitions (33)

  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • proof : Proof of Theorem \ref{['group']}
  • Corollary 3.2
  • proof
  • Corollary 4.1
  • proof
  • ...and 23 more