第3回

日時

2022年7月19日 16:30~18:00(JST)

場所

Zoom

講演者

湯淺 亘氏(大阪公立大学数学研究所/京都大学数理解析研究所)

タイトル

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

アブストラクト

We introduce a skein algebra consisting of $\mathfrak{sp}_4$-webs on a marked surface $\Sigma$ with certain “clasped’’ skein relations at special points and its $\mathbb{Z}_q$-subalgebra (“$\mathbb{Z}_q$-form’’). We prove that its boundary-localization of the $\mathbb{Z}_q$-form is included into a quantum cluster algebra that quantizes the function ring of the moduli space of all decorated twisted $Sp_4$-local systems on $\Sigma$. We also propose a characterization of cluster variables using $\mathfrak{sp}_4$-webs in the spirit of Fomin–Pylyavksyy’s work on $\mathfrak{sl}_3$ (arXiv:1210.1888). Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to our previous work on $\mathfrak{sl}_3$ (arXiv:2101.00643). In this talk, we focus on the construction of the inclusion from the $\mathbb{Z}_q$-form of the $\mathfrak{sp}_4$-skein algebra into the quantum cluster algebra and the characterization of cluster variables. This talk is based on a joint work with Tsukasa Ishibashi (arXiv:2207.01540).

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