Quantum cluster algebra realization for stated ${\mathrm{S}\mathrm{L}}_n$-skein algebras and rotation-invariant bases for polygons

We construct a quantum cluster structure on the skew-field of fractions $\mathrm{F}\mathrm{r}\mathrm{a}\mathrm{c}({\mathcal{S}}_{\omega }(\mathfrak{S}))$ of the stated ${\mathrm{S}\mathrm{L}}_n$-skein algebra ${\mathcal{S}}_{\omega }(\mathfrak{S})$, where $\mathfrak{S}$ is a triangulable pb surface without interior punctures. This work complements the construction for the projected stated skein algebra ${\tilde{\mathcal{S}}}_{\omega }(\mathfrak{S})$ given by the last two authors. Let ${\mathcal{S}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})$ denote the localization of ${\mathcal{S}}_{\omega }(\mathfrak{S})$ at the multiplicative set generated by all frozen variables. Let ${\mathcal{A}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})$ and ${\mathcal{U}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})$ (respectively ${\bar{\mathcal{A}}}_{\omega }(\mathfrak{S})$ and ${\bar{\mathcal{U}}}_{\omega }(\mathfrak{S})$) denote the quantum cluster algebra and quantum upper cluster algebra associated to $\mathrm{F}\mathrm{r}\mathrm{a}\mathrm{c}({\mathcal{S}}_{\omega }(\mathfrak{S}))$ (respectively $\mathrm{F}\mathrm{r}\mathrm{a}\mathrm{c}({\tilde{\mathcal{S}}}_{\omega }(\mathfrak{S}))$). We prove that $${\tilde{\mathcal{S}}}_{\omega }(\mathfrak{S})={\bar{\mathcal{A}}}_{\omega }(\mathfrak{S})={\bar{\mathcal{U}}}_{\omega }(\mathfrak{S})\text{and}{\mathcal{S}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})={\mathcal{A}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})={\mathcal{U}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})$$ whenever $\mathfrak{S}$ is a polygon. As a consequence, when $\mathfrak{S}$ is a polygon, we show that the theta basis of ${\bar{\mathcal{U}}}_{\omega }(\mathfrak{S})$ (respectively ${\mathcal{U}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})$) yields a rotation-invariant basis of ${\bar{\mathcal{S}}}_{\omega }(\mathfrak{S})$ (respectively ${\mathcal{S}}_{\omega }^{\mathrm{f}\mathrm{r}}(\mathfrak{S})$) with several desirable properties, including positivity and a natural parametrization.