Centers and representations of ${\rm SL}_n$ quantum Teichm\"uller spaces

In this paper, we compute the center of the balanced Fock-Goncharov algebra and determine its rank over the center when the quantum parameter is a root of unity. These results have potential applications to the study of the center and rank of the ${\rm SL}_n$-skein algebra. Building on this computation, we classify the irreducible representations of the balanced Fock-Goncharov algebra. Due to the Frobenius homomorphism, every irreducible representation of the (projected) ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$ determines a point in the ${\rm SL}_n$ character variety of $\mathfrak{S}$, known as the classical shadow of the representation. By pulling back the irreducible representations of the balanced Fock-Goncharov algebra via the quantum trace map, we show that there exists a ``large'' subset of the ${\rm SL}_n$ character variety such that, for any point in this subset, there exists an irreducible representation of the (projected) ${\rm SL}_n$-skein algebra whose classical shadow is this point. Finally, we prove that, under mild conditions, the representations of the ${\rm SL}_n$-skein algebra obtained in this way are independent of the choice of ideal triangulation.