Skein and cluster algebras of unpunctured surfaces for sp4

Abstract

As a sequel to our previous work [18] on the ${\mathrm{sl}}_3$-case, we introduce a skein algebra $S_{{\mathrm{sp}}_4,\Sigma }^q$ consisting of ${\mathrm{sp}}_4$-webs on a marked surface Σ, incorporating certain “clasped” skein relations at special points. We further investigate its cluster structure. We also define a natural $Z_q$-form $S_{{\mathrm{sp}}_4,\Sigma }^{Z_q}\subset S_{{\mathrm{sp}}_4,\Sigma }^q$, while the natural coefficient ring $R$ of $S_{{\mathrm{sp}}_4,\Sigma }^q$ includes the inverse of the quantum integer ${\left[2\right]}_q$. We prove that its boundary-localization $S_{{\mathrm{sp}}_4,\Sigma }^{Z_q}\left[{\partial }^{-1}\right]$ embeds into a quantum cluster algebra $A_{{\mathrm{sp}}_4,\Sigma }^q$ that quantizes the function ring of the moduli space $A_{S{p}_4,\Sigma }^{\times }$. Furthermore, we establish the positivity of Laurent expressions of elevation-preserving webs, following an approach similar to [18]. We also propose a characterization of cluster variables in the spirit of Fomin–Pylyavskyy [9] using ${\mathrm{sp}}_4$-webs, and provide infinitely many supporting examples on a quadrilateral.