Do we already know the classification of the finite-dimensional irreducible unitary representations of the modular group $PSL(2,\mathbb{Z})=\mathbb{Z}/2\*\mathbb{Z}/3$?
I'm particularly interested in the following question.


**Q:** Is there a maximal dimension for the finite-dimensional irreducible unitary representations of $PSL(2,\mathbb{Z})$?



Irreducible unitary representation of PSL(2,Z)
The answer is **No**.


There is a mod-$p$ map $f:PSL(2,\mathbb Z) \rightarrow PSL(2,\mathbb F\_p)$.


The permutation representation of $PSL(2,\mathbb F\_p)$ on the projective line on $\mathbb F\_p$ with $p+1$ points splits as a $1$-dimensional and a $p$-dimensional irreducible representation, which can be checked by the [character table](http://www2.math.umd.edu/%7Ejda/characters/psl2/).


Thus, there is a $p$-dimensional irreducible representation for $PSL(2,\mathbb Z)$. As $p$ can be arbitrarily large, there is no maximal dimension for the finite-dimensional irreducible unitary representations of $PSL(2,\mathbb Z) $.


