Cyclotomic factors and irreducibility of the denominators of q-deformed rational numbers
Abstract
For d≥2 the q-deformed modular group specialized at a primitive d-th root of unity is the triangle group of type (2,3,d). Using this we determine the fractions r/s for which the d-th cyclotomic polynomial divides the denominator Sr/s(q) of the q-deformed rational number [r/s]q. They form the orbit of ∞ under the normal closure of the translation z↦ z+d in PSL(2,ℤ). This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences s≡0 and r≡±1 modulo d characterize the divisibility exactly for d≤5. For a∈{2,3,4,6} and n>5a2 prime to a we show that Sa/n(q) is irreducible up to cyclotomic factors. Together with a computer check for small primes, this confirms a conjecture of Kogiso, Ren, Wakui, Yanagawa and the author for every prime p and every r prime to p with r≡± a or ar≡±1pmod p.
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BibTeX
@misc{miyamoto2026cyclotomic,
title = {{Cyclotomic factors and irreducibility of the denominators of $q$-deformed rational numbers}},
author = {Kengo Miyamoto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33703},
eprint = {2609.33703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33703},
}
RIS
TY - UNPB
TI - Cyclotomic factors and irreducibility of the denominators of q-deformed rational numbers
AU - Kengo Miyamoto
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33703
UR - https://arxiv.org/abs/2609.33703
AB - For d≥2 the q-deformed modular group specialized at a primitive d-th root of unity is the triangle group of type (2,3,d). Using this we determine the fractions r/s for which the d-th cyclotomic polynomial divides the denominator S_r/s(q) of the q-deformed rational number [r/s]_q. They form the orbit of ∞ under the normal closure of the translation z↦ z+d in PSL(2,ℤ). This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences s≡0 and r≡±1 modulo d characterize the divisibility exactly for d≤5. For a∈{2,3,4,6} and n>5a^2 prime to a we show that S_a/n(q) is irreducible up to cyclotomic factors. Together with a computer check for small primes, this confirms a conjecture of Kogiso, Ren, Wakui, Yanagawa and the author for every prime p and every r prime to p with r≡± a or ar≡±1pmod p.
ER -