Skein and cluster algebras of unpunctured surfaces for ${\mathfrak{s}\mathfrak{p}}_4$

Continuing to our previous work IY21 on the ${\mathfrak{s}\mathfrak{l}}_3$-case, we introduce a skein algebra ${\mathcal{S}}_{{\mathfrak{s}\mathfrak{p}}_4,\mathrm{\Sigma }}^q$ consisting of ${\mathfrak{s}\mathfrak{p}}_4$-webs on a marked surface $\mathrm{\Sigma }$ with certain “clasped” skein relations at special points, and investigate its cluster nature. We also introduce a natural ${\mathbb{Z}}_q$-form ${\mathcal{S}}_{{\mathfrak{s}\mathfrak{p}}_4,\mathrm{\Sigma }}^{{\mathbb{Z}}_q}\subset {\mathcal{S}}_{{\mathfrak{s}\mathfrak{p}}_4,\mathrm{\Sigma }}^q$, while the natural coefficient ring $\mathcal{R}$ of ${\mathcal{S}}_{{\mathfrak{s}\mathfrak{p}}_4,\mathrm{\Sigma }}^q$ includes the inverse of the quantum integer $[2{]}_q$. We prove that its boundary-localization ${\mathcal{S}}_{{\mathfrak{s}\mathfrak{p}}_4,\mathrm{\Sigma }}^{{\mathbb{Z}}_q}[{\mathrm{\partial }}^{-1}]$ is included into a quantum cluster algebra ${\mathcal{A}}_{{\mathfrak{s}\mathfrak{p}}_4,\mathrm{\Sigma }}^q$ that quantizes the function ring of the moduli space ${\mathcal{A}}_{S{p}_4,\mathrm{\Sigma }}^{\times }$. Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to IY21. We also propose a characterization of cluster variables in the spirit of Fomin–Pylyavksyy FP16 in terms of the ${\mathfrak{s}\mathfrak{p}}_4$-webs, and give infinitely many supporting examples on a quadrilateral.