Frobenius homomorphisms for stated ${\rm SL}_n$-skein modules

TL;DR

This work constructs and analyzes Frobenius homomorphisms for stated ${\rm SL}_n$-skein modules at roots of unity, providing a unified skein-theoretic framework that generalizes the classical $n=2$ case. The main innovation is describing the image of framed knots under the Frobenius map by threading along the knot with reduced power elementary polynomials ${\bar P}_{N,k}$, extending Bonahon–Wong’s result to general $n$ and marking settings. The authors ground the construction in representation theory (Lusztig’s Frobenius, Adams operations, and module-trace maps), and establish compatibility with cutting, quantum traces to tori, and with annular/triangular decompositions, while extending to marked 3-manifolds and projected skein algebras. The results yield structural insights into centers and transparent elements at roots of unity, with potential connections to quantum cluster algebras and skein-Coverings in higher rank cases. Overall, the paper provides a robust, representation-theoretic, and skein-theoretic method to study Frobenius maps and their skein-theoretic consequences for ${\rm SL}_n$ skein theory.

Abstract

The stated ${\rm SL}_n$-skein algebra $\mathscr{S}_{\hat{q}}(\mathfrak{S})$ of a surface $\mathfrak{S}$ is a quantization of the ${\rm SL}_n$-character variety, and is spanned over $\mathbb{Z}[\hat{q}^{\pm 1}]$ by framed tangles in $\mathfrak{S} \times (-1,1)$. If $\hat{q}$ is evaluated at a root of unity $\hatω$ with the order of $\hatω^{4n^2}$ being $N$, then for $\hatη = \hatω^{N^2}$, the Frobenius homomorphism $Φ: \mathscr{S}_{\hatη}(\mathfrak{S}) \to \mathscr{S}_{\hatω}(\mathfrak{S})$ is a surface generalization of the well-known Frobenius homomorphism between quantum groups. We show that the image under $Φ$ of a framed oriented knot $α$ is given by threading along $α$ of the reduced power elementary polynomial, which is an ${\rm SL}_n$-analog of the Chebyshev polynomial $T_N$. This generalizes Bonahon and Wong's result for $n=2$, and confirms a conjecture of Bonahon and Higgins. Our proof uses representation theory of quantum groups and its skein theoretic interpretation, and does not require heavy computations. We also extend our result to marked 3-manifolds.

Figures (16)

• Figure 1: (a) The twice punctured sphere $\mathsf A$ (b) A knot in $\mathsf A$
• Figure 2: (a) & (b) (based) bigon $\mathbb{P}_2$. (c) The stated arc in $\mathbb{P}_2$
• Figure 3: Oriented arcs $b,c,d$ in $\mathbb{P}_3$. We use $b_{i_1j_1}$ to denote the stated arc in $\mathbb{P}_3$ as shown in the picture (the same for $c_{i_2j_2}$ and $d_{i_3j_3}$).
• Figure 4: Monogon vertex and corner arcs
• Figure 5: Barycentric coordinates $(ijk)$ and a $5$-triangulation with its quiver

Theorems & Definitions (111)

• Theorem 1.1: Theorem \ref{['thmFrob']}
• Theorem 1.2: Theorem \ref{['thm11']}
• Theorem 1.3: Theorem \ref{['thm-L-comp-F']}(b)
• Definition 3.1: see KSPW
• Definition 3.2: see PW
• Lemma 3.3
• proof
• Theorem 3.4
• Proposition 4.1
• Lemma 4.2: BH23