The Unicity Theorem and the center of the ${\mathrm{SL}}_3$-skein algebra

Abstract

The ${\mathrm{SL}}_3$-skein algebra ${\mathcal{S}}_{\bar{q}}(\mathfrak{S})$ of a punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${\mathrm{SL}}_3$-character variety of $\mathfrak{S}$. When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of ${\mathcal{S}}_{\bar{q}}(\mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of ${\mathcal{S}}_{\bar{q}}(\mathfrak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for ${\mathcal{S}}_{\bar{q}}(\mathfrak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${\mathrm{SL}}_3$. We compute the rank of ${\mathcal{S}}_{\bar{q}}(\mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation.

v3: 69 pages, streamlined, added some details and a new section 9