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Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominentexamples of cluster $\mathscr{X}$-varieties and higher Teichm\"uller spaces. Ina previous paper of the author (arXiv:2011.14765), the so-called ${\rm SL}_3$quantum trace map is constructed for each triangulable punctured surface$\frak{S}$ and its ideal triangulation $\Delta$, as a homomorphism from thestated ${\rm SL}_3$-skein algebra of the surface to a quantum torus algebrathat deforms the ring of Laurent polynomials in the cube-roots of the clustercoordinate variables for the cluster $\mathscr{X}$-chart for $\mathscr{X}_{{\rmPGL}_3,\frak{S}}$ associated to $\Delta$. We develop quantum mutation mapsbetween special subalgebras of the cube-root quantum torus algebras fordifferent triangulations, and show that the ${\rm SL}_3$ quantum trace maps arecompatible under these quantum mutation maps. As a result, the quantum ${\rmSL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown tobe independent of the choice of an ideal triangulation.

The mutation compatibility of the ${\rm SL}_3$ quantum trace maps for surfaces