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

TL;DR

This paper proves the Unicity Theorem for the ${ m SL}_3$-skein algebra at roots of unity, describes its center, and computes the dimension of generic irreducible representations, advancing understanding of quantum surface invariants.

Contribution

It establishes the Unicity Theorem for ${ m SL}_3$-skein algebras at roots of unity and characterizes the algebra's center, including a surface generalization of Frobenius homomorphisms.

Findings

Proves the Unicity Theorem for irreducible representations.

Shows the center is generated by peripheral skeins and Frobenius images.

Computes the rank of the skein algebra over its center.

Abstract

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.