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

TL;DR

This paper constructs a skein algebra for $ ext{Sp}_4$ webs on surfaces, demonstrating its embedding into a quantum cluster algebra and establishing positivity and a web-based characterization of cluster variables.

Contribution

It introduces a new skein algebra for $ ext{Sp}_4$-webs, proves its inclusion into a quantum cluster algebra, and characterizes cluster variables via webs, extending prior $ ext{sl}_3$ work.

Findings

The skein algebra embeds into a quantum cluster algebra.

Positivity of Laurent expressions for webs is established.

Cluster variables are characterized by $ ext{Sp}_4$-webs.

Abstract

Continuing to our previous work [IY21](arXiv:2101.00643) 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](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of Fomin--Pylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral.