Combinatorics/Computer Science
Invertible substitutions on a three-letter alphabet
Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 111-116.

We study the structure of invertible substitutions on a three-letter alphabet. We show that there exists a finite set $𝕊$ of invertible substitutions such that any invertible substitution can be written as Iwσ1σ2∘⋯∘σk, where Iw is the inner automorphism associated with w, and ${\sigma }_{j}\in 𝕊$ for 1⩽jk. As a consequence, M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.

Nous étudions la structure des substitutions inversibles sur un alphabet à trois lettres. Nous prouvons qu'il existe un ensemble fini $𝕊$ de substitutions inversibles tel que toute substitution inversible puisse être écrite comme Iwσ1σ2∘⋯∘σk, où Iw est l'automorphisme intérieur associé à w et où ${\sigma }_{j}\in 𝕊$ pour 1⩽jk. Comme conséquence, M est la matrice d'une substitution inversible si et seulement si elle est un produit fini de matrices élémentaires non-négatives.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)00006-7

Bo Tan 1; Zhi-Xiong Wen 1; Yiping Zhang 1

1 Department of Mathematics, Wuhan University, 430072 Wuhan, Hubei, PR China
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Bo Tan; Zhi-Xiong Wen; Yiping Zhang. Invertible substitutions on a three-letter alphabet. Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 111-116. doi : 10.1016/S1631-073X(02)00006-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00006-7/

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Research supported by NSFC and by the Special Funds for Major State Basic Research Projects of China.