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

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 $\hat{\omega}$ with the order of $\hat{\omega}^{4n^2}$ being $N$, then for $\hat{\eta} = \hat{\omega}^{N^2}$, the Frobenius homomorphism $\Phi : \mathscr{S}_{\hat{\eta}}(\mathfrak{S}) \to \mathscr{S}_{\hat{\omega}}(\mathfrak{S})$ is a surface generalization of the well-known Frobenius homomorphism between quantum groups. We show that the image under $\Phi$ of a framed oriented knot $\alpha$ is given by threading along $\alpha$ 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.