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