This project is maintained by ysykimura
2022年5月17日 17:00~18:30
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Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates
We introduce and study rational unbounded $\mathfrak{sl}_3$-laminations on a marked surface $\Sigma$, as a topological model for the tropical points of the cluster $\mathcal{X}$-variety associated with the pair $(\mathfrak{sl}_3,\Sigma)$. They are defined as Kuperberg’s $\mathfrak{sl}_3$-webs equipped with certain additional data of rational weights and signs. We introduce their shear coordinates in the same spirit as the Fock—Goncharov’s construction for the $\mathfrak{sl}_2$-case, and show that their coordinate transformations are cluster $\mathcal{X}$-transformations. In the case where $\Sigma$ has no puncture, we also establish a one-to-one correspondence between the integral unbounded $\mathfrak{sl}_3$-laminations (with pinnings) and the basis webs of the $\mathfrak{sl}_3$-skein algebra investigated in my previous work with Wataru Yuasa (arXiv:2101.00643), which should be a basic ingredient for a topological construction of quantum duality map. This talk is based on a joint work with Shunsuke Kano (arXiv:2204.08947).