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

Abstract

We construct a quantum cluster structure on the skew-field of fractions $\mathrm{Frac}({\mathcal{S}}_{\omega }(\mathfrak{S}))$ of the stated ${\mathrm{SL}}_n$-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 ${\mathcal{S}}_{\omega }^{\mathrm{fr}}(\mathfrak{S})$ denote the localization of ${\mathcal{S}}_{\omega }(\mathfrak{S})$ at the multiplicative set generated by all frozen variables. Let ${\mathcal{A}}_{\omega }^{\mathrm{fr}}(\mathfrak{S})$ and ${\mathcal{U}}_{\omega }^{\mathrm{fr}}(\mathfrak{S})$ (respectively ${\overline{\mathcal{A}}}_{\omega }(\mathfrak{S})$ and ${\overline{\mathcal{U}}}_{\omega }(\mathfrak{S})$) denote the quantum cluster algebra and quantum upper cluster algebra associated to $\mathrm{Frac}({\mathcal{S}}_{\omega }(\mathfrak{S}))$ (respectively $\mathrm{Frac}({\tilde{\mathcal{S}}}_{\omega }(\mathfrak{S}))$). We prove that \[ \widetilde{\mathscr S}_ω(\mathfrak{S}) = \overline{\mathscr A}_ω(\mathfrak{S}) = \overline{\mathscr U}_ω(\mathfrak{S}) \quad \text{and} \quad {\mathscr S}_ω^{\rm fr}(\mathfrak{S}) = {\mathscr A}_ω^{\rm fr}(\mathfrak{S}) = {\mathscr U}_ω^{\rm fr}(\mathfrak{S}) \] whenever $\mathfrak{S}$ is a polygon. As a consequence, when $\mathfrak{S}$ is a polygon, we show that the theta basis of ${\overline{\mathcal{U}}}_{\omega }(\mathfrak{S})$ (respectively ${\mathcal{U}}_{\omega }^{\mathrm{fr}}(\mathfrak{S})$) yields a rotation-invariant basis of ${\overline{\mathcal{S}}}_{\omega }(\mathfrak{S})$ (respectively ${\mathcal{S}}_{\omega }^{\mathrm{fr}}(\mathfrak{S})$) with several desirable properties, including positivity and a natural parametrization.

84 pages