The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a longstanding open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A -analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this -analog. The proof is based on the evaluation at . Other roots of unity provide some information on the original problem, which corresponds to the case . We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.
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Keywords: Markoff number, Christoffel word, $q$-analog
Labbé, Sébastien 1; Lapointe, Mélodie 2; Steiner, Wolfgang 3
CC-BY 4.0
@article{ALCO_2023__6_6_1677_0,
author = {Labb\'e, S\'ebastien and Lapointe, M\'elodie and Steiner, Wolfgang},
title = {A $q$-analog of the {Markoff} injectivity conjecture holds},
journal = {Algebraic Combinatorics},
pages = {1677--1685},
publisher = {The Combinatorics Consortium},
volume = {6},
number = {6},
year = {2023},
doi = {10.5802/alco.322},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.322/}
}
TY - JOUR AU - Labbé, Sébastien AU - Lapointe, Mélodie AU - Steiner, Wolfgang TI - A $q$-analog of the Markoff injectivity conjecture holds JO - Algebraic Combinatorics PY - 2023 SP - 1677 EP - 1685 VL - 6 IS - 6 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.322/ DO - 10.5802/alco.322 LA - en ID - ALCO_2023__6_6_1677_0 ER -
%0 Journal Article %A Labbé, Sébastien %A Lapointe, Mélodie %A Steiner, Wolfgang %T A $q$-analog of the Markoff injectivity conjecture holds %J Algebraic Combinatorics %D 2023 %P 1677-1685 %V 6 %N 6 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.322/ %R 10.5802/alco.322 %G en %F ALCO_2023__6_6_1677_0
Labbé, Sébastien; Lapointe, Mélodie; Steiner, Wolfgang. A $q$-analog of the Markoff injectivity conjecture holds. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1677-1685. doi : 10.5802/alco.322. https://alco.centre-mersenne.org/articles/10.5802/alco.322/
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