Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$ (2207.01540v2)

Abstract: Continuing to our previous work IY21 on the $\mathfrak{sl}3$-case, we introduce a skein algebra $\mathscr{S}{\mathfrak{sp}4,\Sigma}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $\Sigma$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$-form $\mathscr{S}{\mathfrak{sp}4,\Sigma}^{\mathbb{Z}_q} \subset \mathscr{S}{\mathfrak{sp}4,\Sigma}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}{\mathfrak{sp}4,\Sigma}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundary-localization $\mathscr{S}{\mathfrak{sp}4,\Sigma}^{\mathbb{Z}_q}[\partial^{-1}]$ is included into a quantum cluster algebra $\mathscr{A}^q{\mathfrak{sp}4,\Sigma}$ that quantizes the function ring of the moduli space $\mathcal{A}{Sp_4,\Sigma}^\times$. 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 $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral.