Arithmetic on $q$-deformed rational numbers
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Kogiso, Takeyoshi Miyamoto, Kengo Ren, Xin Wakui, Michihisa Yanagawa, Kohji |
| author_facet | Kogiso, Takeyoshi Miyamoto, Kengo Ren, Xin Wakui, Michihisa Yanagawa, Kohji |
| contents | Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q) \in {\mathbb Z}[q]$ with $\mathcal{R}_{\frac{r}s}(1)=r,~\mathcal{S}_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{S}_{\frac{r}s}(ζ)=0$ for $ζ=(-1+\sqrt{-3})/2$ (resp. $ζ=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic on $q$-deformed rational numbers Kogiso, Takeyoshi Miyamoto, Kengo Ren, Xin Wakui, Michihisa Yanagawa, Kohji Combinatorics Quantum Algebra 05A30, 11A55, 16G20, 57K14 Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q) \in {\mathbb Z}[q]$ with $\mathcal{R}_{\frac{r}s}(1)=r,~\mathcal{S}_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{S}_{\frac{r}s}(ζ)=0$ for $ζ=(-1+\sqrt{-3})/2$ (resp. $ζ=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links. |
| title | Arithmetic on $q$-deformed rational numbers |
| topic | Combinatorics Quantum Algebra 05A30, 11A55, 16G20, 57K14 |
| url | https://arxiv.org/abs/2403.08446 |