Quantum cluster algebra realization for stated ${\rm SL}_n$-skein algebras and rotation-invariant bases for polygons

Abstract

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