My question probably follows from something basic in character theory, but I don't see what I'm missing. Let $\rho: \Gamma \to \mathrm{GL}(2,\mathbb C)$ be a two-dimensional representation of the modular group $\Gamma$, assumed to be indecomposable. Then for the identity $I$ one has $\rho(-I) = \pm I$. Now I am confused by:
Because $\det \rho(-I) = 1$, $\det \rho$ is a character of order dividing $6$.
I do not see how this follows. I assume here the determinant character means if $\rho$ affords a character $\chi$ then the determinant character $\det\chi$ is defined as $\det \rho(\cdot)$ and here the author means order in the sense as an element of the group of characters.
I have read that the determinant character has order dividing $|G|$, but in this case the modular group $C_2 *C_3$ has infinite cardinality. What am I missing?
Let $\chi$ be a character $\in Hom(SL_2(\Bbb{Z}),\Bbb{C}^\times)$.
Here $\chi(-I)=1$ thus $\chi$ is a character of $SL_2(\Bbb{Z})/\pm I= G=PSL_2(\Bbb{Z})$ whence of $G/[G,G]$.
Then $G= \langle S\rangle \ast \langle ST\rangle \cong C_2 \ast C_3$ (free product of two cyclic groups) so $G/[G,G]\cong C_2 \times C_3$ ie. $\chi$ has order dividing $6$.