Limits of the quantum SO(3) representations for the one-holed torus

Abstract

For $N\ge 2$, we study a certain sequence $({\rho }_p^{(c_p)})$ of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno \cite{1} holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as $p$ tends to infinity. Moreover, the limits describe the action of $S{L}_2(\mathbb{Z})$ on the space of homogeneous polynomials of two variables of total degree $N-1$.