Cyclotomic factors and irreducibility of the denominators of $q$-deformed rational numbers

Abstract

For $d\ge2$ 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 $\infty$ under the normal closure of the translation $z\mapsto z+d$ in $\mathrm{PSL}(2,\mathbb{Z})$. This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences $s\equiv0$ and $r\equiv\pm1$ modulo $d$ characterize the divisibility exactly for $d\le5$. For $a\in\{2,3,4,6\}$ 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 $r\equiv\pm a$ or $ar\equiv\pm1\pmod p$.